THEORY   AND   CALCULATION   OF   TRAN- 
SIENT  ELECTRIC   PHENOMENA 
AND   OSCILLATIONS 


Published   by  the 

McGraw-Hill    Book.  Company 


to  theBookDepariments  of  tKe 

McGraw  Publishing  Company  Hil  1  Publishing1  Company 

Publishers   of  Books  for 

Electrical  World  The  Engineering  and  Mining  Journal 

Engineering  Record  ,  Power  and  The  Engineer 

Electric  Railway  Journal  American   Machinist 

Metallurgical  and  Cnemical  Engineering 


THEOEY  AND  CALCULATION 

OF 

TRANSIENT  ELECTRIC  PHENOMENA 
AND  OSCILLATIONS 


BY 

CHARLES  PROTEUS  STEINMETZ 


FIRST  EDITION 
SECOND  IMPRESSION  CORRECTED 


McGRAW-HILL   BOOK   COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,   LONDON,   B.C. 
1909 


LIBRARY 

UNIVERSITY  OF  CALIFORNIA 
DAVIS 


COPYRIGHT,  1909, 

BY    THE 

McGRAW   PUBLISHING   COMPANY 
NEW   YORK 


Stanhope  Jpresa 

F.    H.   GILSON     COMPANY 
BOSTON,     U.S.A. 


DEDICATED 

TO    THE 

MEMORY  OF   MY  FRIEND   AND   TEACHER 
KUDOLF  EICKEMEYEB, 


PREFACE 


THE  following  work  owes  its  origin  to  a  course  of  instruction 
given  during  the  last  few  years  to  the  senior  class  in  electrical 
engineering  at  Union  University  and  represents  the  work  of  a 
number  of  years.  It  comprises  the  investigation  of  phenomena 
which  heretofore  have  rarely  been  dealt  with  in  text-books  but 
have  now  become  of  such  importance  that  a  knowledge  of  them 
is  essential  for  every  electrical  engineer,  as  they  include  some  of 
the  most  important  problems  which  electrical  engineering  will 
have  to  solve  in  the  near  future  to  maintain  its  thus  far  unbroken 
progress. 

A  few  of  these  transient  phenomena  were  observed  and  experi- 
mentally investigated  in  the  early  days  of  electrical  engineering, 
for  instance,  the  building  up  of  the  voltage  of  direct-current 
generators  from  the  remanent  magnetism.  Others,  such  as  the 
investigation  of  the  rapidity  of  the  response  of  a  compound 
generator  or  a  booster  to  a  change  of  load,  have  become  of  impor- 
tance with  the  stricter  requirements  now  made  on  electric  systems. 
Transient  phenomena  which  were  of  such  short  duration  and 
small  magnitude  as  to  be  negligible  with  the  small  apparatus  of 
former  days  have  become  of  serious  importance  in  the  huge 
generators  and  high  power  systems  of  to-day,  as  the  discharge  of 
generator  fields,  the  starting  currents  of  transformers,  the  short- 
circuit  currents  of  alternators,  etc.  Especially  is  this  the  case 
with  two  classes  of  phenomena  closely  related  to  each  other:  the 
phenomena  of  distributed  capacity  and  those  of  high  frequency 
currents.  Formerly  high  frequency  currents  were  only  a  subject 
for  brilliant  lecture  experiments;  now,  however,  in  the  wireless 
telegraphy  they  have  found  an  important  industrial  use.  Teleph- 
ony has  advanced  from  the  art  of  designing  elaborate  switch- 
boards to  an  engineering  science,  due  to  the  work  of  M.  I.  Pupin 

vii 


PREFACE  TO  THE   SECOND  EDITION 


DUE  to  the  relatively  short  time  which  has  elapsed  since 
the  appearance  of  the  first  edition,  no  material  changes  or 
additions  were  needed  in  the  preparation  of  the  second  edition. 
The  work  has  been  carefully  perused  and  typographical  and 
other  errors,  which  had  passed  into  the  first  edition,  were 
eliminated.  In  this,  thanks  are  due  to  those  readers  who 
have  drawn  my  attention  to  errors. 

Since  the  appearance  of  the  first  edition,  the  industrial 
importance  of  transients  has  materially  increased,  and  con- 
siderable attention  has  thus  been  devoted  to  them  by  engineers. 
The  term  " transient"  has  thereby  found  an  introduction,  as 
noun,  into  the  technical  language,  instead  of  the  more  cumber- 
some expression  ''transient  phenomenon,"  and  the  former  term 
is  therefore  used  to  some  extent  in  the  revised  edition. 

As  appendix  have  been  added  tables  of  the  velocity  functions 
of  the  electric  field,  sil  x  and  col  x,  and  similar  functions, 
together  with  explanation  of  their  mathematical  relations,  as 
tables  of  these  functions  are  necessary  in  calculations  of  wave 
propagation,  but  are  otherwise  difficult  to  get.  These  tables 
were  derived  from  tables  of  related  functions  published  by 
J.  W.  L.  Glaisher,  Philosophical  Transactions  of  the  Royal 
Society  of  London,  1870,  Vol.  160. 


CONTENTS. 


SECTION  I.     TRANSIENTS  IN  TIME. 

PAGB 

CHAPTER  I.     THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT.  3 

1.  Flow    of    electric    energy,    the    electric    field    and    its 

components.  3 

2.  The  electromagnetic  field,  the  electrostatic  field  and  the 

power  consumption,  and  their  relation  to  current  and 
voltage.  5 

3.  The  electromagnetic  energy,  the  electrostatic  energy,  and 

the  power  loss  of  the  circuit,  and  their  relations  to  the 
circuit  constants,  inductance,  capacity  and  resistance.  6 

4.  Effect  of  conductor  shape  and  material  on  resistance, 

inductance  and  capacity.  8 

5.  The  resistance  of  materials :  metals,  electrolytes,  insulators 

and  pyroelectrolytes.  8 

6.  Inductance  and  the  magnetic  characteristics  of  materials. 

Permeability  and  saturation,  and  its  effect  on  the  mag- 
netic field  of  the  circuit.  9 

7.  Capacity  and  the  dielectric  constant  of  materials.     The 

disruptive  strength  of  materials,  and  its  effect  on  the 
electrostatic  field  of  the  circuit.  11 

8.  Power  consumption    in    changing    magnetic   and    static 

fields:  magnetic  and  dielectric  hysteresis.  Effective 
resistance  and  shunted  conductance.  12 

9.  Magnitude  of  resistance,  inductance  and  capacity  in  in- 

dustrial circuits.     Circuits  of  negligible  capacity.  12 

10.  Gradual  change  of  circuit  conditions  in  a  circuit  of  negli- 

gible capacity.  Effect  of  capacity  in  allowing  a  sudden 
change  of  circuit  conditions,  causing  a  surge  of  energy 
between  magnetic  and  static.  14 

CHAPTER  II.     INTRODUCTION.  16 

11.  The  usual  equations  of  electric  circuit  do  not  apply  to  the 

time  immediately  after  a  circuit  changes,  but  a  transient 
term  then  appears.  16 

12.  Example  of  the  transient  term  in  closing  or  opening  a  con- 

tinuous current  circuit :  the  building  up  and  the  dying 
out  of  the  direct  current  in  an  alternator  field.  16 

xi 


xii  CONTENTS. 

PAGE 

13.  Example  of  transient  term  produced  by  capacity:  the 

charge  and  discharge  of  a  condenser,  through  an  induc- 
tive circuit.  Conditions  for  oscillations,  and  the  possi- 
bility of  excessive  currents  and  voltages.  17 

14.  Example  of  the  gradual  and  the  oscillatory  approach  of 

an  alternating  current  to  its  permanent  value.  20 

15.  Conditions  for  appearance  of  transient  terms,  and  for 

their  harmlessness  or  danger.     Effect  of  capacity.  21 

16.  Relations  of  transient  terms  and  their  character  to  the 

stored  energy  of  the  circuit.  21 

17.  Recurrent  or  periodic  transient  terms:  their  appearance  in 

rectification.  22 

18.  Oscillating  arcs  and  arcing  ground  of  transmission  line, 

as  an  example  of  recurrent  transient  terms.  22 

19.  Cases  in  which  transient  phenomena  are  of  industrial  im- 

portance. 23 

CHAPTER    III.   INDUCTANCE    AND    RESISTANCE    IN    CONTINUOUS- 
CURRENT  CIRCUITS.  25 

20.  Equations   of   continuous-current    circuit,    including   its 

transient  term.  25 

21.  Example  of  a  continuous-current  motor  circuit.  27 

22.  Excitation  of  a  motor  field.'    Time  required   for  shunt 

motor  field  to  build  up  or  discharge.  Conditions  of 
design  to  secure  quick  response  of  field.  27 

23.  Discharge  of  shunt  motor  field  while  the  motor  is  coming 

to  rest.     Numerical  example.  29 

24.  Self -excitation  of  direct-current  generator:  the  effect  of 

the  magnetic  saturation  curve.  Derivation  of  the 
general  equations  of  the  building  up  of  the  shunt 
generator.  Calculations  of  numerical  example.  32 

25.  Self -excitation  of  direct-current  series  machine.     Numeri- 

cal example  of  time  required  by  railway  motor  to  build 

up  as  generator  or  brake.  38 


CHAPTER    IV.   INDUCTANCE    AND    RESISTANCE    IN    ALTERNATING- 
CURRENT  CIRCUITS.  41 

26.  Derivation  of  general  equations,  including  transient  term.       41 

27.  Conditions  for  maximum  value,  and  of  disappearance  of 

transient  term.     Numerical  examples;  lighting  circuit, 
motor  circuit,  transformer  and  reactive  coil.  43 

28.  Graphic  representation  of  transient  term.  45 


CONTENTS.  xiii 

PAGE 

CHAPTER  V.   RESISTANCE,  INDUCTANCE  AND  CAPACITY  IN  SERIES. 

CONDENSER  CHARGE  AND  DISCHARGE.  47 

29.  The  differential  equations  of  condenser  charge  and  dis- 

charge. 47 

30.  Integration  of  these  equations.  48 

31.  Final  equations  of   condenser  charge   and  discharge,  in 

exponential  form.  50 

32.  Numerical  example.  51 

33.  The  three  cases  of  condenser  charge  and  discharge :  loga- 

rithmic, critical  and  oscillatory.  52 

34.  The  logarithmic  case,  and  the  effect  of  resistance  in  elimi- 

nating excessive  voltages  in  condenser  discharges.  53 

35.  Condenser  discharge  in  a  non-inductive  circuit.  54 

36.  Condenser  charge  and  discharge  in  a  circuit  of  very  small 

inductance,  discussion  thereof,  and  numerical  example.       55 

37.  Equations  of  the  critical  case  of  condenser  charge  and  dis- 

charge.    Discussion.  56 

38.  Numerical  example.  58 

39.  Trigonometric    or   oscillatory   case.     Derivation   of   the 

equations  of  the  condenser  oscillation.  Oscillatory  con- 
denser charge  and  discharge.  58 

40.  Numerical  example.  61 

41.  Oscillating  waves  of  current  and  e.m.f.  produced  by  con- 

denser discharge.  Their  general  equations  and  frequen- 
cies. 62 

42.  High  frequency  oscillations,  and  their  equations.  63 

43.  The  decrement  of  the  oscillating  wave.  The  effect  of  resist- 

ance on  the  damping,  and  the  critical  resistance. 
Numerical  example.  65 

CHAPTER  VI.     OSCILLATING  CURRENTS.  67 

44.  Limitation  of  frequency  of  alternating  currents  by  genera- 

tor design ;  limitation  of  usefulness  of  oscillating  current 

by  damping  due  to  resistance.  67 

45.  Discussion  of  sizes  of  inductances  and  capacities,  and  their 

rating  in  kilovolt-amperes.  68 

46.  Condenser  discharge  equations,  discussion  and  design.  69 

47.  Condenser  discharge  efficiency  and  damping.  71 

48.  Independence  of  oscillating  current  frequency  on  size  of 

condenser  and  inductance.  Limitations  of  frequency 
by  mechanical  size  and  power.  Highest  available 
frequencies.  .  72 


xiv  CONTENTS. 

PACE 

49.  The  oscillating  current  generator,  discussion  of  its  design.  74 

50.  The  equations  of  the  oscillating  current  generator.  76 

51.  Discussion  of  equations:  frequency,  current,  power,  ratio 

of  transformation.  79 

52.  Calculation  of  numerical  example  of  a  generator  having  a 

frequency  of  hundreds  of  thousands  of  cycles  per  second.  82 

53.  52  Continued.  86 

54.  Example  of  underground  cable  acting  as  oscillating  cur- 

rent generator  of  low  frequency.  87 

CHAPTER  VII.     RESISTANCE,  INDUCTANCE  AND  CAPACITY  IN  SERIES 

IN  ALTERNATING  CURRENT  CIRCUIT.  88 

55.  Derivation  of  the  general  equations.     Exponential  form.  88 

56.  Critical  case.  .    92 

57.  Trigonometric  or  oscillatory  case.  93 

58.  Numerical  example.  94 

59.  Oscillating  start  of  alternating  current  circuit.  96 

60.  Discussion  of  the  conditions  of  its  occurrence.  98 

61.  Examples.  100 

62.  Discussion  of  the  application  of  the  equations  to  trans- 

mission lines  and  high-potential  cable  circuits.  102 

63.  The  physical  meaning  and  origin  of  the  transient  term.  103 

CHAPTER   VIII.     LOW-FREQUENCY  'SURGES    IN    HIGH-POTENTIAL 

SYSTEMS.  105 

64.  Discussion  of  high  potential  oscillations  in  transmission 

lines  and  underground  cables.  105 

65.  Derivation  of  the  equations  of  current  and  condenser 

potentials  and  their  components.  106 

66.  Maximum  and  minimum  values  of  oscillation.  109 

67.  Opening  the  circuit  of  a  transmission  line  under  load.  112 

68.  Rupturing  a  short-circuit  of  a  transmission  line.  113 

69.  Numerical    example  of  starting  transmission  line  at  no 

load,  opening  it  at  full  load,  and  opening  short-circuit.  116 

70.  Numerical  example  of  a  short-circuit  oscillation  of  under- 

ground cable  system.  119 

71.  Conclusions.  120 

CHAPTER  IX.     DIVIDED  CIRCUIT.  121 

72.  General  equations  of  a  divided  circuit.  121 

73.  Resolution  into  permanent  term  and  transient  term.  124 

74.  Equations  of  special  case  of  divided  continuous-current 

circuit  without  capacity.  126 


CONTENTS.  XV 

PAGE 

75.  Numerical  example  of  a  divided  circuit  having  a  low- 

resistance  inductive,  and  a  high-resistance  noninduc- 
tive  branch.  129 

76.  Discussion  of  the  transient  term  in  divided  circuits,  and 

its  industrial  use.  130 

77.  Example  of  the  effect  of  a  current  pulsation  in  a  circuit  on 

a  voltmeter  shunting  an  inductive  part  of  the  circuit.  131 

78.  Capacity  shunting  a  part  of  the  continuous-current  circuit. 

Derivation  of  equations.  133 

79.  Calculations  of  numerical  example.  136 

80.  Discussions  of  the  elimination  of  current  pulsations  by 

shunted  capacity.  137 

81.  Example  of  elimination  of  pulsation  from  non-inductive 

circuit,  by  shunted  capacity  and  series  inductance.  139 

CHAPTER  X.     MUTUAL  INDUCTANCE.  141 

82.  The   differential   equations   of   mutually   inductive    cir- 

cuits. 141 

83.  Their  discussion.  143 

84.  Circuits   containing  resistance,   inductance   and   mutual 

inductance,  but  no  capacity.  144 

85.  Integration  of  their  differential  equations,  and  their  dis- 

cussion. 146 

86.  Case  of  constant  impressed  e.m.fs.  147 

87.  The  building  up  (or  down)  of  an  over-compounded  direct- 

current  generator,  at  sudden  changes  of  load.  149 

88.  87  Continued.  152 

89.  87  Continued.  154 

90.  Excitation  of  series  booster,  with  solid  and  laminated 

field  poles.     Calculation  of  eddy  currents  in  solid  field 
iron.  155 

91.  The  response  of  a  series  booster  to  sudden  change  of 

load.  158 

92.  Mutual  inductance  in  circuits  containing  self-inductance 

and  capacity.     Integration  of  the  differential  equations.     161 

93.  Example :  the  equations  of  the  Ruhmkorff  coil  or  induc- 

torium.  164 

94.  93  Continued.  166 

CHAPTER  XI.     GENERAL  SYSTEM  OF  CIRCUITS.  168 

95.  Circuits  containing  resistance  and  inductance  only.  168 

96.  Application  to  an  example.  171 


xvi  CONTENTS. 

PAGE 

97.  Circuit  containing  resistance,  self  and  mutual  inductance 

and  capacity.  174 

98.  Discussion  of  the  general  solution  of  the  problem.  177 


CHAPTER  XII.     MAGNETIC  SATURATION  AND  HYSTERESIS  IN  MAG- 
NETIC CIRCUITS.  179 

99.   The  transient  term  in  a  circuit  of  constant  inductance.         179 

100.  Variation  of  inductance  by  magnetic  saturation  causing 

excessive  transient  currents.  180 

101.  Magnetic  cycle  causing  indeterminate  values  of  transient 

currents.  181 

102.  Effect  of  frequency  on  transient  terms  to  be  expected  in 

transformers.  181 

103.  Effect  of  magnetic  stray  field  or  leakage  on  transient 

starting  current  of  transformer.  182 

104.  Effect  of  the  resistance,  equations,  and  method  of  con- 

struction of  transient  current  of  transformer  when 
starting.  185 

105.  Construction  of  numerical  examples,  by  table.  188 

CHAPTER  XIII.     TRANSIENT  TERM  OF  THE  ROTATING  FIELD.  191 

106.  Polyphase  m.m.fs.  producing  magnetic  field  of  constant 

intensity,  revolving  with  uniform  synchronous  veloc- 
ity. 191 

107.  The  sum  of  instantaneous  values  of  the  permanent  as 

well  as  the  transient  term  of  polyphase  m.m.fs.  equals 
zero.  191 

108.  Equation   of  the   resultant   of   a   system   of  polyphase 

m.m.fs.,  in  any  direction,  its  permanent  and  its  transient 
term.  Maximum  value  of  permanent  term.  Nu- 
merical* example.  192 

109.  Direction    of    maximum    intensity    of    transient    term. 

Velocity  of  its  rotation.     Oscillating  character  of  it.         194 

110.  Intensity  of  maximum  value.     Its  lag  and  lead  behind 

uniform  rotation.     Numerical  example.  196 

111.  Discussion.     Independence  of  transient  term  on  phase 

angle  at  start.  197 

CHAPTER  XIV.     SHORT-CIRCUIT  CURRENTS  OP  ALTERNATORS.  199 

112.  Relation  of  permanent  short-circuit  current  to  armature 

reaction  and  self-inductance.  Value  of  permanent 
short-circuit  current.  \99 


CONTENTS.  ivii 

PAGE 

113.  Relation  of  momentary  short-circuit  current  to  arma- 

ture reaction  and  self-inductance.     Value  of  momen- 
tary short-circuit  current.  200 

114.  Transient  term  of  revolving  field  of  armature  reaction. 

Pulsating  armature  reaction  of  single-phase  alternator.     201 

115.  Polyphase  alternator.     Calculation  of  field  current  during 

short-circuit.     Equivalent  reactance  of  armature  reac- 
tion.    Self-inductance  in  field  circuit.  204 

116.  Equations  of  armature  short-circuit  current  and  short- 

circuit  armature  reaction.  207 

117.  Numerical  example.  208 

118.  Single-phase  alternator.     Calculation   of  pulsating  field 

current  at  short-circuit.  209 

119.  Equations  of  armature  short-circuit  current  and  short- 

circuit  armature  reaction.  210 

120.  Numerical  example.  212 


SECTION  II.     PERIODIC   TRANSIENTS. 

CHAPTER  I.     INTRODUCTION.  217 

1.  General    character   of   periodically  recurring   transient 

phenomena  in  time.  217 

2.  Periodic  transient  phenomena  with  single  cycle.  218 

3.  Multi-cycle  periodic  transient  phenomena.  218 

4.  Industrial  importance  of  periodic  transient  phenomena: 

circuit  control,  high  frequency  generation,  rectification.  220 

5.  Types  of  rectifiers.     Arc  machines.  221 

CHAPTER  II.  CIRCUIT  CONTROL  BY  PERIODIC  TRANSIENT  PHENOM- 
ENA. 223 

6.  Tirrill  Regulator.  223 

7.  Equations.  224 

8.  Amplitude  of  pulsation.  226 

CHAPTER  III.     MECHANICAL  RECTIFICATION.  229 

9.  Phenomena  during  reversal,  and  types  of  mechanical  rec- 

tifiers. 229 

10.  Single-phase  constant-current  rectification:  compounding 

of  alternators  by  rectification.  231 

11.  Example  and  numerical  calculations.  233 

12.  Single-phase   constant-potential  rectification:  equations.  236 


xviii  CONTENTS. 

PAGE 

13.  Special  case,  calculation  of  numerical  example.  239 

14.  Quarter-phase     rectification:     Brush     arc     machine. 

Equations.  242 

15.  Calculation  of  example.  246 

CHAPTER  IV.     ARC  RECTIFICATION.  249 

16.  The  rectifying  character  of  the  arc.  249 

17.  Mercury  arc  rectifier.     Constant-potential  and  constant- 

current  type.  250 

18.  Mode  of  operation  of  mercury  arc  rectifier:    Angle  of 

over-lap.  252 

19.  Constant-current   rectifier:  Arrangement  of  apparatus.  255 

20.  Theory  and  calculation:  Differential  equations.  256 

21.  Integral  equations.  258 

22.  Terminal  conditions  and  final  equations.  260 

23.  Calculation  of  numerical  example.  262 

24.  Performance  curves  and  oscillograms.     Transient  term.  263 

25.  Equivalent  sine  waves:  their  derivation.  267 

26.  25  Continued.  269 

27.  Equations  of  the  equivalent  sine  waves  of  the  mercury  arc 

rectifier.     Numerical  example.  271 

SECTION  III.    TRANSIENTS  IN   SPACE. 

CHAPTER  I.     INTRODUCTION.  277 

1.  Transient  phenomena  in  space,  as  periodic  functions  of 

time  and  transient  functions  of  distance,  represented  by 

transient  functions  of  complex  variables.  277 

2.  Industrial  importance  of  transient  phenomena  in  space.  278 

CHAPTER  II.     LONG  DISTANCE  TRANSMISSION  LINE.  279 

3.  Relation  of  wave  length  of  impressed  frequency  to  natural 

frequency  of  line,  and  limits  of  approximate  line  cal- 
culations. 279 

4.  Electrical  and  magnetic  phenomena  in  transmission  line.  281 

5.  The  four  constants  of  the  transmission  line :  r,  L,  g,  C.  282 

6.  The  problem  of  the  transmission  line.  283 

7.  The  differential  equations  of  the  transmission  line,  and 

their  integral  equations. 

8.  Different  forms  of  the  transmission  line  equations.  287 

9.  Equations  with  current  and  voltage  given  at  one  end  of 

the  line.  289 
10.   Equations  with  generator  voltage,  and  load  on  receiving 

circuit  given.  291 


CONTENTS.  xix 

PAGE 

11.  Example  of  60,000-volt  200-mile  line.  292 

12.  Comparison  of  result  with  different  approximate  calcula- 

tions. 294 

13.  Wave  length  and  phase  angle.  295 

14.  Zero  phase  angle  and  45-degree  phase  angle.     Cable  of 

negligible  inductance.  296 

15.  Examples  of  non-inductive,  lagging  and  leading  load,  and 

discussion  of  flow  of  energy.  297 

16.  Special  case:  Open  circuit  at  end  of  line.  299 

17.  Special  case:  Line  grounded  at  end.  304 

18.  Special  case :  Infinitely  long  conductor.  305 

19.  Special  case:  Generator  feeding  into  closed  circuit.  306 

20.  Special  case:  Line  of  quarter-wave  length,  of  negligible 

resistance.  306 

21.  Line  of  quarter-wave  length,  containing  resistance  r  and 

conductance  g.  309 

22.  Constant-potential  —  constant-current  transformation  by 

line  of  quarter-wave  length.  310 

23.  Example  of  excessive  voltage  produced  in  high-potential 

transformer  coil  as  quarter- wave  circuit.  312 

24.  Effect  of  quarter-wave  phenomena  on  regulation  of  long 

transmission  lines;  quarter-wave  transmission.  313 

25.  Limitations  of  quarter-wave  transmission.  314 

26.  Example  of  quarter-wave  transmission  of  60,000  kw.  at  60 

cycles,  over  700  miles.  315 

CHAPTER  III.    THE  NATURAL  PERIOD  OF  THE  TRANSMISSION  LINE.  320 

27.  The  oscillation  of  the  transmission  line  as  condenser.  320 

28.  The  conditions  of  free  oscillation.  321 

29.  Circuit  open  at  one  end,  grounded  at  other  end.  322 

30.  Quarter-wave  oscillation  of  transmission  line.  323 

31.  Frequencies  of  line  discharges,  and  complex  discharge 

wave.  327 

32.  Example  of  discharge  of  line  of  constant  voltage  and  zero 

current.  329 

33.  Example  of  short-circuit  oscillation  of  line.  331 

34.  Circuit  grounded  at  both  ends :  Half-wave  oscillation.  333 

35.  The  even  harmonics  of  the  half-wave  oscillation.  334 

36.  Circuit  open  at  both  ends.  335 

37.  Circuit  closed  upon  itself:  Full-wave  oscillation.  336 

38.  Wave  shape  and  frequency  of  oscillation.  338 

39.  Time  decrement  of  oscillation,  and  energy  transfer  be- 

tween sections  of  complex  oscillating  circuit.  339 


xx  CONTENTS. 

PAGE 

CHAPTER  IV.     DISTRIBUTED  CAPACITY  OF  HIGH-POTENTIAL  TRANS- 
FORMER. 342 

40.  The  transformer  coil  as  circuit  of  distributed  capacity,  and 

the  character  of  its  capacity.  342 

41.  The  differential  equations  of  the  transformer  coil,  and 

their  integral  equations.  344 

42.  Terminal  conditions  and  final  approximate  equations.        346 

CHAPTER  V.   DISTRIBUTED  SERIES  CAPACITY.  348 

43.  Potential  distribution  in  multigap  circuit.  348 

44.  Probable  relation  of  the  multigap  circuit  to  the  lightning 

flash  in  the  clouds.  349 

45.  The  differential  equations  of  the  multigap  circuit,  and 

their  integral  equations.  350 

46.  Terminal  conditions,  and  final  equations.  351 

47.  Numerical  example.  353 

CHAPTER  VI.     ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION.  355 

48.  Magnetic  screening  by  secondary  currents  in  alternating 

fields.  355 

49.  The  differential  equations  of  alternating  magnetic  flux 

in  a  lamina.  356 

50.  Their  integral  equations.  357 

51.  Terminal  conditions,  and  the  final  equations.  358 

52.  Equations  for  very  thick  laminae.  360 

53.  Wave  length,  attenuation,  depth  of  penetration.  361 

54.  Numerical   example,   with   frequencies  of  60,   1000  and 

10,000  cycles  per  second.  362 

55.  Depth  of  penetration  of  alternating  magnetic   flux  in 

different  metals.  363 

56.  Wave  length,  attenuation,  and  velocity  of  penetration.         365 

57.  Apparent  permeability,  as  function  of  frequency,  and 

damping.  366 

58.  Numerical  example  and  discussion.  367 

CHAPTER  VII.     DISTRIBUTION  OF  ALTERNATING-CURRENT  DENSITY 

IN  CONDUCTOR.  369 

59.  Cause  and  effect  of  unequal  current  distribution.     In- 

dustrial importance.  369 

60.  Subdivision  and   stranding.     Flat   conductor  and  large 

conductor.  371 


CONTENTS.  xxi 

PAGE 

61.  The  differential  equations  of  alternating-current  distri- 

bution in  a  flat  conductor.  374 

62.  Their  integral  equations.  375 

63.  Mean  value  of  current,  and  effective  resistance.  376 

64.  Equations  for  large  conductors.  377 

65.  Effective  resistance  and  depth  of  penetration.  379 

66.  Depth  of  penetration,  or  conducting  layer,  for  different 

materials   and    different   frequencies,   and    maximum 
economical  conductor  diameter.  384 

CHAPTER  VIII.     VELOCITY  OF  PROPAGATION  OP  ELECTRIC  FIELD.     387 

67.  Conditions,  under  which  the  velocity  of  propagation  of 

the  field  is  of  industrial  importance.  387 

68.  Equations   of   decrease   of    electric   field   with   the   dis- 

tance. 388 

69.  Effect   of  return   conductor  on  distance   decrement   of 

field.  389 

70.  Inductance  of  length  I  of  infinitely  long  conductor  with- 

out return  conductor.  390 

71.  Equations  of  magnetic  flux,  effective  resistance  of  radia- 

tion, inductance  and  impedance.  391 

72.  Evaluation  of  functions  sil  al  and  col  al.  394 

73.  Self-inductive  impedance,  and  numerical  example.  395 

74.  Discussion  of  effective  resistance  and  radiated  power,  as 

function  of  frequency.  396 

75.  Mutual  inductance  of  two  distant  conductors  of  finite 

length.  398 

76.  Example.  399 

77.  Capacity  of  a  sphere  in  space.  400 

78.  Example.  401 

79.  Sphere  at  a  distance  from  ground.  402 

CHAPTER  IX.     HIGH-FREQUENCY  CONDUCTORS.  403 

80.  Effect  of  the  frequency  on  the  constants  of  a  conductor.     403 

81.  Thermal  resistance  and  radiation  resistance,  internal  and 

external  reactance,  as  functions  of  the  frequency.  405 

82.  Total  impedance  of  high  frequency  conductor,  and  its 

components,  discussion.  407 

83.  Example  of  copper  and  iron  wire,  copper  ribbon  and  iron 

pipe ;  tabulation  and  discussion  of  numerical  values.  408 

84.  Continued  discussion  of  results.  •  409 

85.  Potential  drop  in  conductors  carrying  high  frequency 

currents.     Tabulation.     Effect  of  conductor  shape  and 
material.  412 


CONTENTS. 

SECTION  IV.     TRANSIENTS  IN   TIME  AND   SPACE. 

PAGE 
CHAPTER  I.     GENERAL  EQUATIONS.  417 

1.  The  constants  of  the  electric  circuit,  and  their  constancy.     417 

2.  The  differential   equations   of   the   general   circuit,   and 

their  general  integral  equations.  419 

3.  Terminal  conditions.     Velocity  of  propagation.  421 

4.  The  group  of  terms  in  the  general  integral  equations 

and  the  relations  between  its  constants.  422 

5.  Elimination  of  the  complex  exponent  in  the  group  equa- 

tions. 425 

6.  Final  form  of  the  general  equations  of  the  electric  circuit.     428 

CHAPTER  II.     DISCUSSION  OF  GENERAL  EQUATIONS.  431 

7.  The   two   component  waves  and   their  reflected  waves. 

Attenuation  in  time  and  in  space.  431 

8.  Period,    wave    length,    time    and    distance    attenuation 

constants.  433 

9.  Simplification  of  equations  at  high  frequency,  and  the 

velocity  unit  of  distance.  434 

10.  Decrement  of  traveling  wave.  436 

11.  Physical  meaning  of  the  two  component  waves.  437 

12.  Stationary  or  standing  wave.   Trigonometric  arid  logarith- 

mic waves.  438 

13.  Propagation  constant  of  wave.  440 

CHAPTER  III.     STANDING  WAVES.  442 

14.  Oscillatory,  critical  and  gradual  standing  wave.  442 

15.  The  wave   length  which  divides  the  gradual  from  the 

oscillatory  wave.  446 

16.  High-power  high-potential    overhead   transmission   line. 

Character  of  waves.      Numerical   example.      General 
equations.  449 

17.  High-potential  underground  power  cable.     Character  of 

waves.     Numerical  example.     General  equations.  452 

18.  Submarine   telegraph   cable.      Existence   of    logarithmic 

waves.  454 

19.  Long  distance   telephone   circuit.     Numerical  example. 

Effect  of  leakage.     Effect  of  inductance  or  "loading."    454 

CHAPTER  IV.    TRAVELING  WAVES.  457 

20.  Different  forms  of  the  equations  of  the  traveling  wave.          457 


CONTENTS.  xxiii 

PAGE 

21.  Component  waves  and  single  traveling  wave.     Attenua- 

tion. 459 

22.  Effect  of  inductance,  as  loading,  and  leakage,  on  attenu- 

ation.    Numerical  example  of  telephone  circuit.  462 

23.  Traveling  sine  wave  and  traveling  cosine  wave.     Ampli- 

tude and  wave  front.  464 

24.  Discussion  of  traveling  wave  as  function  of  distance,  and 

of  time.  466 

25.  Numerical  example,  and  its  discussion.  469 

26.  The   alternating-current   long-distance  line  equations  as 

special  case  of  a  traveling  wave.  471 

27.  Reduction  of  the  general  equations  of  the  special  traveling 

wave  to  the  standard  form  of  alternating-current  trans- 
mission line  equations.  474 

CHAPTER  V.     FREE  OSCILLATIONS.  478 

28.  Types  of  waves:  standing  waves,  traveling  waves,  alter- 

nating-current waves.  478 

29.  Conditions  and  types  of  free  oscillations.  478 

30.  Terminal  conditions.  480 

31.  Free  oscillation  as  standing  wave.  481 

32.  Quarter-wave  and  half-wave  oscillation,  and  their  equa- 

tions. 482 

33.  Conditions  under  which  a  standing  wave  is  a  free  oscilla- 

tion, and  the  power  nodes  of  the  free  oscillation.  485 

34.  Wave  length,  and  angular  measure  of  distance.  487 

35.  Equations  of  quarter-wave  and  half-wave  oscillation.  489 

36.  Terminal  conditions.     Distribution  of  current  and  voltage 

at  start,  and  evaluation  of  the  coefficients  of  the  trigo- 
nometric series.  491 

37.  Final  equations  of  quarter-wave  and  half- wave  oscilla- 

tion. 492 

38.  Numerical  example  of  the  discharge  of  a  transmission  line.     493 

39.  Numerical  example  of  the  discharge  of  a  live  line  into  a 

dead  line.  496 

CHAPTER  VI.     TRANSITION  POINTS  AND  THE  COMPLEX  CIRCUIT.        498 

40.  General  discussion.  498 

41.  Transformation  of  general  equations,  to  velocity  unit  of 

distance.  499 

42.  Discussion.  501 

43.  Relations  between  constants,  at  transition  point.  502 


xxiv  CONTENTS. 

PAGE 

44.  The  general  equations  of  the  complex  circuit,  and  the 

resultant  time  decrement %  503 

45.  Equations    between   integration   constants  of  adjoining 

sections.  504 

46.  The  energy  transfer  constant  of  the  circuit  section,  and 

the  transfer  of  power  between  the  sections.  507 

47.  The  final  form  of  the  general  equations  of  the  complex 

circuit,  .     508 

48.  Full-wave,  half-wave,  quarter-wave  oscillation,  and  gen- 

eral high-frequency  oscillation.  509 

49.  Determination  of  the  resultant  time  decrement  of  the  cir- 

cuit. 510 

CHAPTER  VII.     POWER  AND  ENERGY  OF  THE  COMPLEX  CIRCUIT.         513 

50.  Instantaneous  power.     Effective  or  mean  power.     Power 

transferred.  513 

51.  Instantaneous  and  effective  value  of  energy  stored  in  the 

magnetic  field ;  its  motion  along  the  circuit,  and  varia- 
tion with  distance  and  with  time.  513 

52.  The  energy  stored  in  the  electrostatic  field  and  its  compo- 

nents.    Transfer  of  energy  between  electrostatic  and 
electromagnetic  field.  517 

53.  Energy  stored  in  a  circuit  section  by  the  total  electric 

field,  and  power  supplied  to  the  circuit  by  it.  518 

54.  Power  dissipated  in  the  resistance  and  the  conductance  of 

a  circuit  section.  519 

55.  Relations  between  power  supplied  by  the  electric  field 

of  a  circuit  section,  power  dissipated  in  it,  and  power 

transferred  to,  or  received  by  other  sections.  520 

56.  Flow  of  energy,  and  resultant  circuit  decrement.  521 

57.  Numerical  examples.  522 

CHAPTER  VIII.    REFLECTION  AND  REFRACTION    AT   TRANSITION 
POINT. 

58.  Main  wave,  reflected  wave  and  transmitted  wave.  525 

59.  Transition  of  single  wave,  constancy  of  phase  angles, 

relations  between  the  components,  and  voltage  trans- 
formation at  transition  point.  526 

60.  Numerical  example,  and  conditions  of  maximum.  530 

61.  Equations  of  reverse  wave.  531 

62.  Equations  of  compound  wave  at  transition  point,  and  its 

three  components.  532 

63.  Distance  phase  angle,  and  the  law  of  refraction.  533 


CONTENTS.  xxv 

PAGE 
CHAPTER  IX.     INDUCTIVE  DISCHARGES.  535 

64.  Massed  inductance  discharging  into  distributed  circuit. 

Combination  of  generating  station  and  transmission 
line.  535 

65.  Equations  of  inductance,   and   change   of   constants  at 

transition  point.  536 

66.  Line  open  or  grounded  at  end.     Evaluation  of  frequency 

constant  and  resultant  decrement.  538 

67.  The  final  equations,  and  their  discussion.  540 

68.  Numerical  example.      Calculation  of  the  first  six  har- 

monics. 542 

APPENDIX:    VELOCITY  FUNCTIONS  OF  THE  ELECTRIC  FIELD.  545 


SECTION   I 
TRANSIENTS   IN   TIME 


TKANSIENTS   IN   TIME 


CHAPTER   I. 

THE   CONSTANTS   OF   THE   ELECTRIC   CIRCUIT. 

1.  To  transmit  electric  energy  from  one  place  where  it  is 
generated  to  another  place  where  it  is  used,  an  electric  cir- 
cuit is  required,  consisting  of  conductors  which  connect  the 
point  of  generation  with  the  point  of  utilization. 

When  electric  energy  flows  through  a  circuit,  phenomena 
take  place  inside  of  the  conductor  as  well  as  in  the  space  out- 
side of  the  conductor. 

In  the  conductor,  during  the  flow  of  electric  energy  through 
the  circuit,  electric  energy  is  consumed  continuously  by  being 
converted  into  heat.  Along  the  circuit,  from  the  generator 
to  the  receiver  circuit,  the  flow  of  energy  steadily  decreases 
by  the  amount  consumed  in  the  conductor,  and  a  power  gradi- 
ent exists  in  the  circuit  along  or  parallel  with  the  conductor. 

(Thus,  while  the  voltage  may  decrease  from  generator  to 
receiver  circuit,  as  is  usually  the  case,  or  may  increase,  as  in 
an  alternating-current  circuit  with  leading  current,  and  while 
the  current  may  remain  constant  throughout  the  circuit,  or 
decrease,  as  in  a  transmission  line  of  considerable  capacity 
with  a  leading  or  non-inductive  receiver  circuit,  the  flow  of 
energy  always  decreases  from  generating  to  receiving  circuit, 
and  the  power  gradient  therefore  is  characteristic  of  the  direc- 
tion of  the  flow  of  energy.) 

In  the  space  outside  of  the  conductor,  during  the  flow  of 
energy  through  the  circuit,  a  condition  of  stress  exists  which 
is  called  the  electric  field  of  the  conductor.  That  is,  the 
surrounding  space  is  not  uniform,  but  has  different  electric 
and  magnetic  properties  in  different  directions. 

No  power  is  required  to  maintain  the  electric  field,  but  energy 

3 


4  TRANSIENT  PHENOMENA 

is  required  to  produce  the  electric  field,  and  this  energy  is 
returned,  more  or  less  completely,  when  the  electric  field  dis- 
appears by  the  stoppage  of  the  flow  of  energy. 

Thus,  in  starting  the  flow  of  electric  energy,  before  a  perma- 
nent condition  is  reached,  a  finite  time  must  elapse  during 
which  the  energy  of  the  electric  field  is  stored,  and  the  generator 
therefore  gives  more  power  than  consumed  in  the  conductor 
and  delivered  at  the  receiving  end;  again,  the  flow  of  electric 
energy  cannot  be  stopped  instantly,  but  first  the  energy  stored 
in  the  electric  field  has  to  be  expended.  As  result  hereof, 
where  the  flow  of  electric  energy  pulsates,  as  in  an  alternating- 
current  circuit,  continuously  electric  energy  is  stored  in  the 
field  during  a  rise  of  the  power,  and  returned  to  the  circuit 
again  during  a  decrease  of  the  power. 

The  electric  field  of  the  conductor  exerts  magnetic  and  elec- 
trostatic actions. 

The  magnetic  action  is  a  maximum  in  the  direction  concen- 
tric, or  approximately  so,  to  the  conductor.  That  is,  a  needle- 
shaped  magnetizable  body,  as  an  iron  needle,  tends  to  set  itself 
in  a  direction  concentric  to  the  conductor. 

The  electrostatic  action  has  a  maximum  in  a  direction  radial, 
or  approximately  so,  to  the  conductor.  That  is,  a  light  needle- 
shaped  conducting  body,  if  the  electrostatic  component  of  the 
field  is  powerful  enough,  tends  to  set  itself  in  a  direction  radial 
to  the  conductor,  and  light  bodies  are  attracted  or  repelled 
radially  to  the  conductor. 

Thus,  the  electric  field  of  a  circuit  over  which  energy  flows 
has  three  main  axes  which  are  at  right  angles  with  each  other: 

The  electromagnetic  axis,  concentric  with  the  conductor. 

The  electrostatic  axis,  radial  to  the  conductor. 

The  power  gradient,  parallel  to  the  conductor. 

This  is  frequently  expressed  pictorially  by  saying  that  the 
lines  of  magnetic  force  of  the  circuit  are  concentric,  the  lines 
of  electrostatic  force  radial  to  the  conductor. 

Where,  as  is  usually  the  case,  the  electric  circuit  consists  of 
several  conductors,  the  electric  fields  of  the  conductors  super- 
impose upon  each  other,  and  the  resultant  lines  of  magnetic 
and  of  electrostatic  forces  are  not  concentric  and  radial  respec- 
tively, except  approximately  in  the  immediate  neighborhood 
of  the  conductor. 


THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT  5 

In  the  electric  field  between  parallel  conductors  the  magnetic 
and  the  electrostatic  lines  of  force  are  conjugate  pencils  of  circles. 

2.  Neither  the  power  consumption  in  the  conductor,  nor 
the  electromagnetic  field,  nor  the  electrostatic  field,  are  pro- 
portional to  the  flow  of  energy  through  the  circuit. 

The  product,  however,  of  the  intensity  of  the  magnetic  field, 
<f>,  and  the  intensity  of  the  electrostatic  field,  "^  is  proportional 
to  the  flow  of  energy  or  the  power,  P,  and  the  power  P  is  there- 
fore resolved  into  a  product  of  two  components,  i  and  e,  which 
are  chosen  proportional  respectively  to  the  intensity  of  the 
magnetic  field  <I>  and  of  the  electrostatic  field  V. 

That  is,  putting 

P  =  ie  (1) 

we  have 

<E>  =  Li  =  the  intensity  of  the  electromagnetic  field.        (2) 
Mf  =  Ce  =  the  intensity  of  the  electrostatic  field.  (3) 

The  component  i,  called  the  current,  is  defined  as  that  factor 
of  the  electric  power  P  which  is  proportional  to  the  magnetic 
field,  and  the  other  component  e,  called  the  voltage,  is  defined 
as  that  factor  of  the  electric  power  P  which  is  proportional  to 
the  electrostatic  field. 

Current  i  and  voltage  e,  therefore,  are  mathematical  fictions, 
factors  of  the  power  P,  introduced  to  represent  respectively  the 
magnetic  and  the  electrostatic  or  "  dielectric  "  phenomena. 

The  current  i  is  measured  by  the  magnetic  action  of  a  circuit, 
as  in  the  ammeter;  the  voltage  e,  by  the  electrostatic  action  of 
a  circuit,  as  in  the  electrostatic  voltmeter,  or  by  producing  a 
current  i  by  the  voltage  e  and  measuring  this  current  i  by  its 
magnetic  action,  in  the  usual  voltmeter. 

The  coefficients  L  and  (7,  which  are  the  proportionality  factors 
of  the  magnetic  and  of  the  dielectric  component  of  the  electric 
field,  are  called  the  inductance  and  the  capacity  of  the  circuit, 
respectively. 

As  electric  power  P  is  resolved  into  the  product  of  current  i 
and  voltage  e,  the  power  loss  in  the  conductor,  Ph  therefore  can 
also  be  resolved  into  a  product  of  current  i  and  voltage  et 
which  is  consumed  in  the  conductor.  That  is, 

P,  =  iet. 


6  TRANSIENT  PHENOMENA 

It  is  found  that  the  voltage  consumed  in  the  conductor,  eh  is 
proportional  to  the  factor  i  of  the  power  P,  that  is, 

et  =  ri,  (4) 

where  r  is  the  proportionality  factor  of  the  voltage  consumed 
by  the  loss  of  power  in  the  conductor,  or  by  the  power  gradient, 
and  is  called  the  resistance  of  the  circuit. 

Any  electric  circuit  therefore  must  have  three  constants, 
r,  L,  and  (7,  where 

r  =  circuit  constant  representing  the  power  gradient,  or  the 
loss  of  power  in  the  conductor,  called  resistance. 

L  =  circuit  constant  representing  the  intensity  of  the  electro- 
magnetic component  of  the  electric  field  of  the  circuit, 
called  inductance. 

C  =  circuit  constant  representing  the  intensity  of  the  electro- 
static component  of  the  electric  field  of  the  circuit,  called 
capacity. 

3.  A  change  of  the  magnetic  field  of  the  conductor,  that  is, 
of  the  number  of  lines  of  magnetic  force  </>  surrounding  the 
conductor,  generates  an  e.m.f. 

'-3  <•> 

in  the  conductor  and  thus  absorbs  a  power 

*"-*-<£  (6) 

or,  by  equation  (2)  :  <J>  =  Li  by  definition,  thus  : 

d&       ,  di         i      r>/       T  •  di 

-  =  L-,and:   P'  =  Lt-,  (7) 

and  the  total  energy  absorbed  by  the  magnetic  field  during  the 
rise  of  current  from  zero  to  i  is 


WM  --P'dt  (8) 

=  LJidi, 


that  is,  ,, 


WM  -  (9) 


THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT  1 

A  change  of  the  dielectric  field  of  the  conductor,  ^,  absorbs 
a  current  proportional  to  the  change  of  the  dielectric  field  : 


and  absorbs  the  power 
or,  by  equation  (3), 


P»=ei'=e—,  (11) 


(12) 


and  the  total  energy  absorbed  by  the  dielectric  field  during  a 
rise  of  voltage  from  0  to  6  is 


WK  ==p"dt  (13) 

=  cfede, 


that  is  *n 


The  power  consumed  in  the  conductor  by  its  resistance  r  is 

Pr  =  ieh  (15) 

and  thus,  by  equation  (4), 

Pr  =  tV.  (16) 

That  is,  when  the  electric  power 

P  =  ei      (1) 
exists  in  a  circuit,  it  is 
pr  =  tfr  =  power  lost  in  the  conductor,      (16) 

WM  =  l—  =  energy  stored  in  the  magnetic  field  of  the  circuit,  (9) 


l 

Ll 


W  K  =  —  =  energy  stored  in  the  dielectric  field  of   the  cir- 
£t 

cuit,     (14) 


8  TRANSIENT  PHENOMENA 

and  the  three  circuit  constants  r,  L,  C  therefore  appear  as  the 
components  of  the  energy  conversion  into  heat,  magnetism,  and 
electric  stress,  respectively,  in  the  circuit. 

4.  The  circuit  constant,  resistance  r,  depends  only  on  the 
size  and  material  of  the  conductor,  but  not  on  the  position  of 
the  conductor  in  space,  nor  on  the  material  filling  the  space 
surrounding  the  conductor,  nor  on  the  shape  of  the  conductor 
section. 

The  circuit  constants,  inductance  L  and  capacity  (7,  almost 
entirely  depend  on  the  position  of  the  conductor  in  space,  on 
the  material  filling  the  space  surrounding  the  conductor,  and 
on  the  shape  of  the  conductor  section,  but  do  not  depend  on 
the  material  of  the  conductor,  except  to  that  small  extent  as 
represented  by  the  electric  field  inside  of  the  conductor  section. 

5.  The  resistance  r  is  proportional  to  the  length  and  inversely 
proportional  to  the  section  of  the  conductor, 

r  =  plT'  (IV) 

J\. 

where  p  is  a  constant  of  the  material,  called  the  resistivity  or 
specific  resistance. 

For  different  materials,  p  varies  probably  over  a  far  greater 
range  than  almost  any  other  physical  quantity.  Given  in  ohms 
per  centimeter  cube,*  it  is,  approximately,  at  ordinary  tem- 
peratures : 

Metals:  Cu  ..................  ...........  1  .  6  X  lO"6 

Al.  ..,  .........................  2.5  X  10-6 

Fe  ..............................  10  X  10-6 

Hg  .............................  94  X  10-6 

Gray  cast  iron  ..............  up  to  100  X  10~6 

High-resistance  alloys  .......  up  to  150  X  10~8 

Electrolytes:  N03H  .............  down  to  1  .3  at  30  per  cent 

KOH  .............  down  to  1  .  9  at  25  per  cent 

NaCL  .  ............  down  to  4  .  7  at  25  per  cent 

up  to 

Pure  river  water  .......................  .  104 

and  over  alcohols,  oils,  etc.,  to  practically  infinity. 

Meaning  a  conductor  of  one  centimeter  length  and  one  square  centimeter 


* 
section. 


THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT  9 

So-called  ' '  insulators"  : 

Fiber about  1012 

Paraffin  oil about  IQ» 

Paraffin about  1014  to  1016 

Mica about  1014 

G1ass about  1014  to  1016 

Rubber about  1016 

Air practically  oo 

In  the  wide  gap  between  the  highest  resistivity  of  metal 
alloys,  about  p  =  150  X  10~6,  and  the  lowest  resistivity  of 
electrolytes,  about  p  =  1,  are 

Carbon:  metallic down  to  100  X  10~6 

amorphous  (dense) 0.04  and  higher 

anthracite very  high 

Silicon  and  Silicon  Alloys: 

Cast  silicon 1  down  to  0 . 04 

Ferro  silicon 0.04  down  to  50  X  10~a 

The  resistivity  of  arcs  and  of  Geissler  tube  discharges  is  of  about 
the  same  magnitude  as  electrolytic  resistivity. 

The  resistivity,  p,  is  usually  a  function  of  the  temperature, 
rising  slightly  with  increase  of  temperature  in  metallic  conduct- 
ors and  decreasing  in  electrolytic  conductors.  Only  with  few 
materials,  as  silicon,  the  temperature  variation  of  p  is  so  enor- 
mous that  p  can  no  longer  be  c.onsidered  as  even  approximately 
constant  for  all  currents  i  which  give  a  considerable  tempera- 
ture rise  in  the  conductor.  Such  materials  are  commonly 
called  pyroelectrolytes. 

6.  The  inductance  L  is  proportional  to  the  section  and 
inversely  proportional  to  the  length  of  the  magnetic  circuit 
surrounding  the  conductor,  and  so  can  be  represented  by 

L  =  ^  (18) 

where  /*  is  a  constant  of  the  material  filling  the  space  surround- 
ing the  conductor,  which  is  called  the  magnetic  permeability. 

As  in  general  neither  section  nor  length  is  constant  in  differ- 
ent parts  of  the  magnetic  circuit  surrounding  an  electric  con- 


10  TRANSIENT  PHENOMENA 

ductor,  the  magnetic  circuit  has  as  a  rule  to  be  calculated 
piecemeal,  or  by  integration  over  the  space  occupied  by  it. 

The  permeability,  /*,  is  constant  and  equals  unity  or  very 
closely  fj.  =  1  for  all  substances,  with  the  exception  of  a  few 
materials  which  are  called  the  magnetic  materials,  as  iron, 
cobalt,  nickel,  etc.,  in  which  it  is  very  much  higher,  reaching 
sometimes  and  under  certain  conditions  in  iron  values  as  high 
as  fjL  =  6000. 

In  these  magnetic  materials  the  permeability  /*  is  not  con- 
stant but  varies  with  the  magnetic  flux  density,  or  number  of 
lines  of  magnetic  force  per  unit  section,  (B,  decreasing  rapidly 
for  high  values  of  (B. 

In  such  materials  the  use  of  the  term  /*  is  therefore  incon- 
venient, and  the  inductance,  L,  is  calculated  by  the  relation 
between  the  magnetizing  force  as  given  in  ampere-turns  per 
unit  length  of  magnetic  circuit,  or  by  "field  intensity,"  and 
magnetic  induction  (B. 

The  magnetic  induction  (B  in  magnetic  materials  is  the  sum 
of  the  "space  induction"  3C,  corresponding  to  unit  permeability, 
plus  the  "metallic  induction"  (B',  which  latter  reaches  a  finite 
limiting  value.  That  is, 

(B  =  3C  +  (&'.  (19) 

The  limiting  values,  or  so-called  "saturation  values,"  of  (B' 
are  approximately,  in  lines  of  magnetic  force  per  square  centi- 
meter: 

Iron .20,000 

Cobalt 12,000- 

Nickel 6,000 

Magnetite 5,000 

Manganese  alloys up  to  4,000 

The  inductance,  L,  therefore  is  a  constant  of  the  circuit  if 
the  space  surrounding  the  conductor  contains  no  magnetic 
material,  and  is  more  or  less  variable  with  the  current,  i,  if 
magnetic  material  exists  in  the  space  surrounding  the  conductor. 
In  the  latter  case,  with  increasing  current,  i,  the  inductance,  L, 
first  slightly  increases,  reaches  a  maximum,  and  then  decreases, 
approaching  as  limiting  value  the  value  which  it  would  have  in 
the  absence  of  the  magnetic  material. 


THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT  11 

7.  The  capacity,  C,  is  proportional  to  the  section  and  inversely 
proportional  to  the  length  of  the  electrostatic  field  of  the  con- 
ductor: 

^      *A 

G  =  T,  (20) 

where  K  is  a  constant  of  the  material  filling  the  space  surround- 
ing the  conductor,  which  is  called  the  "dielectric  constant,"  or 
the  " specific  capacity/'  or  "  permittivity." 

Usually  the  section  and  the  length  of  the  different  parts  of 
the  electrostatic  circuit  are  different,  and  the  capacity  therefore 
has  to  be  calculated  piecemeal,  or  by  integration. 

The  dielectric  constant  K  of  different  materials  varies  over  a 
relative  narrow  range  only.  It  is  approximately: 

AC  =  1  in  the  vacuum,  in  air  and  in  other  gases, 
K  =  2  to  3  in  oils,  paraffins,  fiber,  etc., 
K  =  3  to  4  in  rubber  and  gutta-percha, 
K  =  3  to  5  in  glass,  mica,  etc., 

reaching  values  as  high  as  7  to  8  in  organic  compounds  of  heavy 
metals,  as  lead  stearate,  and  about  12  in  sulphur. 

The  dielectric  constant,  /c,  is  practically  constant  for  all  voltages 
e,  up  to  that  voltage  at  which  the  electrostatic  field  intensity, 
or  the  electrostatic  gradient,  that  is,  the  "volts  per  centimeter," 
exceeds  a  certain  value  d,  which  depends  upon  the  material  and 
which  is  called  the  "dielectric  strength"  or  "disruptive  strength" 
of  the  material.  At  this  potential  gradient  the  medium  breaks 
down  mechanically,  by  puncture,  and  ceases  to  insulate,  but 
electricity  passes  and  so  equalizes  the  potential  gradient. 

The  disruptive  strength,  d,  given  in  volts  per  centimeter  is 
approximately : 

Air:  32,000. 

Oils:  250,000  to  1,000,000. 

Mica:  up  to  4,000,000. 

The  capacity,  (7,  of  a  circuit  therefore  is  constant  up  to  the 
voltage  e,  at  which  at  some  place  of  the  electrostatic  field  the 
dielectric  strength  is  exceeded,  disruption  takes  place,  and  a 
part  of  the  surrounding  space  therefore  is  made  conducting,  and 
by  this  increase  of  the  effective  size  of  the  conductor  the  capacity 
C  is  increased. 


12  TRANSIENT  PHENOMENA 

8.  Of  the  amount  of  energy  consumed  in  creating  the  electric 
field  of  the  circuit  not  all  is  returned  at  the  disappearance  of 
the  electric  field,  but  a  part  is  consumed  by  conversion  into  heat 
in  producing  or  in  any  other  way  changing  the  electric  field. 
That  is,  the  conversion  of  electric  energy  into  and  from  the 
electromagnetic  and  electrostatic  stress  is  not  complete,  but  a 
loss  of  energy  occurs,  especially  with  the  magnetic  field  in  the 
so-called  magnetic  materials,  and  with  the  electrostatic  field  in 
unhomogeneous  dielectrics. 

The  energy  loss  in  the  production  and  reconversion  of  the 
magnetic  component  of  the  field  can  be  represented  by  an 
effective  resistance  rr  which  adds  itself  to  the  resistance  r0  of 
the  conductor  and  more  or  less  increases  it. 

The  energy  loss  in  the  electrostatic  field  can  be  represented 
by  an  effective  resistance  r" ,  shunting  across  the  circuit,  and 
consuming  an  energy  current  i" ,  in  addition  to  the  current  i  in 
the  conductor.  Usually,  instead  of  an  effective  resistance  r", 
its  reciprocal  is  used,  that  is,  the  energy  loss  in  the  electro- 
static field  represented  by  a  shunted  conductance  g. 

In  its  most  general  form  the  electric  circuit  therefore  contains 
the  constants : 

1.   Inductance  L,  storing 'the  energy,  -— , 

ft 


2.  Capacity  C,  storing  the  energy,  -  —  > 

& 

3.  Resistance  r  =  r0  +  r',  consuming  the  power,  tfr  =  ? 

4.  Conductance  g}  consuming  the  power,  ezg, 

where  r0  is  the  resistance  of  the  conductor,  r'  the  effective  resist- 
ance representing  the  power  loss  in  the  magnetic  field  L,  and  g 
represents  the  power  loss  in  the  electrostatic  field  C. 

9.  If  of  the  three  components  of  the  electric  field,  the  electro- 
magnetic stress,  electrostatic  stress,  and  the  power  gradient,  one 
equals  zero,  a  second  one  must  equal  zero  also.  That  is,  either 
all  of  the  three  components  exist  or  only  one  exists. 

Electric  systems  in  which  the  magnetic  component  of  the 
field  is  absent,  while  the  electrostatic  component  may  be  consider- 
able, are  represented  for  instance  by  an  electric  generator  or 
a  battery  on  open  circuit,  or  by  the  electrostatic  machine.  In 
such  systems  the  disruptive  effects  due  to  high  voltage,  there- 


THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT  13 

fore,  are  most  pronounced,  while  the  power  is  negligible,  and 
phenomena  of  this  character  are  usually  called  " static." 

Electric  systems  in  which  the  electrostatic  component  of  the 
field  is  absent,  while  the  electromagnetic  component  is  consider- 
able, are  represented  for  instance  by  the  short-circuited  secondary 
coil  of  a  transformer,  in  which  no  potential  difference  and,  there- 
fore, no  electrostatic  field  exists,  since  the  generated  e.m.f.  is 
consumed  at  the  place  of  generation.  Practically  negligible  also 
is  the  electrostatic  component  in  all  low-voltage  circuits. 

The  effect  of  the  resistance  on  the  flow  of  electric  energy  in 
industrial  applications  is  restricted  to  fairly  narrow  limits:  as 
the  resistance  of  the  circuit  consumes  power  and  thus  lowers  the 
efficiency  of  the  electric  transmission,  it  is  uneconomical  to 
permit  too  high  a  resistance.  As  lower  resistance  requires  a 
larger  expenditure  of  conductor  material,  it  is  usually  uneco- 
nomical to  lower  the  resistance  of  the  circuit  below  that  which, 
gives  a  reasonable  efficiency. 

As  result  hereof,  practically  always  the  relative  resistance, 
that  is,  the  ratio  of  the  power  lost  in  the  resistance  to  the  total 
power,  lies  between  2  per  cent  and  20  per  cent. 

It  is  different  with  the  inductance  L  and  the  capacity  C.     Of 

the  two  forms  of  stored  energy,  the  magnetic  —  and  electro- 

e*C 

static ,  usually  one  is  so  small  that  it  can  be  neglected  com- 

fk 

pared  with  the  other,  and  the  electric  circuit  with  sufficient 
approximation  treated  as  containing  resistance  and  inductance, 
or  resistance  and  capacity  only. 

In  the  so-called  electrostatic  machine  and  its  applications, 
frequently  only  capacity  and  resistance  come  into  consideration. 

In  all  lighting  and  power  distribution  circuits,  direct  current 
or  alternating  current,  as  the  110-  and  220-volt  lighting  circuits, 
the  500-volt  railway  circuits,  the  2000-volt  primary  distribution 
circuits,  due  to  the  relatively  low  voltage,  the  electrostatic 

energy  -  ^  is  still  so  very  small  compared   with  the  electro- 

2 

magnetic  energy,  that  the  capacity  C  can  for  most  purposes  be 
neglected  and  the  circuit  treated  as  containing  resistance  and 
inductance  only. 


14  TRANSIENT  PHENOMENA 

Of  approximately  equal  magnitude  is  the  electromagnetic 

energy  ——  and  the  electrostatic  energy  -  ^  in  the  high-potential 

Zi  iL 

long-distance  transmission  circuit,  in  the  telephone  circuit,  and 
in  the  condenser  discharge,  and  so  in  most  of  the  phenomena 
resulting  from  lightning  or  other  disturbances.  In  these  cases 
all  three  circuit  constants,  r,  L,  and  C,  are  of  essential  impor- 
tance. 

10.  In  an  electric  circuit  of  negligible  inductance  L  and 
negligible  capacity  C,  no  energy  is  stored,  and  a  change  in  the 
circuit  thus  can  be  brought  about  instantly  without  any  disturb- 
ance or  intermediary  transient  condition. 

In  a  circuit  containing  only  resistance  and "  capacity ,  as  a 
static  machine,  or  only  resistance  arid  inductance,  as  a  low  or 
medium  voltage  power  circuit,  electric  energy  is  stored  essentially 
in  one  form  only,  and  a  change  of  the  circuit,  as  an  opening  of 
the  circuit,  thus  cannot  be  brought  about  instantly,  but  occurs 
more  or  less  gradually,  as  the  energy  first  has  to  be  stored  or 
discharged. 

In  a  circuit  containing  resistance,  inductance,  and  capacity, 
and  therefore  capable  of  storing  energy  in  two  different  forms, 
the  mechanical  change  of  circuit  conditions,  as  the  opening  of  a 
circuit,  can  be  brought  about  instantly,  the  internal  energy  of 
the  circuit  adjusting  itself  to  the  changed  circuit  conditions  by 
a  transfer  of  energy  between  static  and  magnetic  and  inversely, 
that  is,  after  the  circuit  conditions  have  been  changed,  a  transient 
phenomenon,  usually  of  oscillatory  nature,  occurs  in  the  circuit 
by  the  readjustment  of  the  stored  energy. 

These  transient  phenomena  of  the  readjustment  of  stored 
electric  energy  with  a  change  of  circuit  conditions  require  careful 
study  wherever  the  amount  of  stored  energy  is  sufficiently  large 
to  cause  serious  damage.  This  is  analogous  to  the  phenomena 
of  the  readjustment  of  the  stored  energy  of  mechanical  motion: 
while  it  may  be  harmless  to  instantly  stop  a  slowly  moving  light 
carriage,  the  instant  stoppage,  as  by  collision,  of  a  fast  railway 
train  leads  to  the  usual  disastrous  result.  So  also,  in  electric 
systems  of  small  stored  energy,  a  sudden  change  of  circuit  con- 
ditions may  be  safe,  while  in  a  high-potential  power  system  of 
very  great  stored  electric  energy  any  change  of  circuit  conditions 
requiring  a  sudden  change  of  energy  is  liable  to  be  destructive. 


THE  CONSTANTS  OF  THE  ELECTRIC  CIRCUIT  15 

Where  electric  energy  is  stored  in  one  form  only,  usually  little 
danger  exists,  since  the  circuit  protects  itself  against  sudden 
change  by  the  energy  adjustment  retarding  the  change,  and 
only  where  energy  is  stored  electrostatically  and  magnetically, 
the  mechanical  change  of  the  circuit  conditions,  as  the  opening 
of  the  circuit,  can  be  brought  about  instantly,  and  the  stored 
energy  then  surges  between  electrostatic  and  magnetic  energy. 

In  the  following,  first  the  phenomena  will  be  considered  which 
result  from  the  stored  energy  and  its  readjustment  in  circuits 
storing  energy  in  one  form  only,  which  usually  is  as  electro- 
magnetic energy,  and  then  the  general  problem  of  a  circuit 
storing  energy  electromagnetically  and  electrostatically  will  be 
considered. 


CHAPTER  II. 

INTRODUCTION. 

11.  In  the  investigation  of  electrical  phenomena,  currents 
and  potential  differences,  whether  continuous  or  alternating, 
are  usually  treated  as  stationary  phenomena.    That  is,   the 
assumption  is  made  that  after  establishing  the  circuit  a  sufficient 
time  has  elapsed  for  the  currents  and  potential  differences  to 
reach  their  final  or  permanent  values,  that  is,  become  constant, 
with  continuous  current,  or  constant  periodic  functions  of  time, 
with  alternating  current.     In  the  first  moment,  however,  after 
establishing  the  circuit,  the  currents  and  potential  differences 
in  the  circuit  have  not  yet  reached  their  permanent  values, 
that  is,  the  electrical  conditions  of  the  circuit  are  not  yet  the 
normal  or  permanent  ones,  but  a  certain  time  elapses  while  the 
electrical  conditions  adjust  themselves. 

12.  For  instance,  a  continuous  e.m.f.,  eOJ  impressed  upon  a 
circuit  of  resistance  r,  produces  and  maintains  in  the  circuit  a 
current, 


In  the  moment  of  closing  the  circuit  of  e.m.f.  e0  on  resistance  r, 
the  current  in  the  circuit  is  zero.  Hence,  after  closing  the  circuit 
the  current  i  has  to  rise  from  zero  to  its  final  value  i0.  If  the 
circuit  contained  only  resistance  but  no  inductance,  this  would 
take  place  instantly,  that  is,  there  would  be  no  transition  period. 
Every  circuit,  however,  contains  some  inductance.  The  induc- 
tance L  of  the  circuit  means  L  interlinkages  of  the  circuit  with 
lines  of  magnetic  force  produced  by  unit  current  in  the  circuit, 
or  iL  interlinkages  by  current  i.  That  is,  in  establishing  current 
i0  in  the  circuit,  the  magnetic  flux  iQL  must  be  produced.  A 
change  of  the  magnetic  flux  iL  surrounding  a  circuit  generates 
in  the  circuit  an  e.m.f., 

d 

e  -  * 

16 


INTRODUCTION 


17 


This  opposes  the  impressed  e.m.f.  e0,  and  therefore  lowers  the 
e.m.f.  available  to  produce  the  current,  and  thereby  the  current, 
which  then  cannot  instantly  assume  its  final  value,  but  rises 
thereto  gradually,  and  so  between  the  starting  of  the  circuit 
and  the  establishment  of  permanent  condition  a  transition 
period  appears.  In  the  same  manner  and  for  the  same  reasons, 
if  the  impressed  e.m.f.  eQ  is  withdrawn,  but  the  circuit  left  closed, 
the  current  i  does  not  instantly  disappear  but  gradually  dies 
out,  as  shown  in  Fig.  1,  which  gives  the  rise  and  the  decay  of  a 


^ 

,__—  • 

IS 

1 

/ 

\ 

5 

-2- 

KM 

?/i 

1 

/ 

1  1 

\ 

I,  = 

• 

01 

en 

r.v- 

1° 

\ 

I 

a 

a 

\ 

1 

< 

\ 

/ 

V 

/ 

\ 

/ 

\ 

s 

'  —  , 

-*^. 

-0- 

( 

1 

I 

I 

, 

5 

! 

i 

j 

; 

L 

I 

1 

! 

I 

5 

Seconds 


Fig.  1.     Rise  and  decay  of  continuous  current  in  an  inductive  circuit. 

continuous  current  in  an  inductive  circuit:  the  exciting  current 
of  an  alternator  field,  or  a  circuit  having  the  constants  r  =  12 
ohms;  L  =  6  henrys,  and  eQ  =  240  volts;  the  abscissas  being 
seconds  of  time. 

13.  If  an  electrostatic  condenser  of  capacity  C  is  connected 
to  a  continuous  e.m.f.  e0,  no  current  exists,  in  stationary  con- 
dition, in  this  direct-current  circuit  (except  that  a  very  small 
current  may  leak  through  the  insulation  or  the  dielectric  of  the 
condenser),  but  the  condenser  is  charged  to  the  potential  dif- 
ference eo;  or  contains  the  electrostatic  charge 

Q  =  to0. 

In  the  moment  of  closing  the  circuit  of  e.m.f.  e0  upon  the 
capacity  C,  the  condenser  contains  no  charge,  that  is,  zero 
potential  difference  exists  at  the  condenser  terminals.  If  there 
were  no  resistance  and  no  inductance  in  the  circuit  in  the 


18  TRANSIENT  PHENOMENA 

moment  of  closing  the  circuit,  an  infinite  current  would  exist 
charging  the  condenser  instantly  to  the  potential  difference  e0. 
If  r  is  the  resistance  of  the  direct-current  circuit  containing  the 
condenser,  and  this  circuit  contains  no  inductance,  the  current 

Q 

starts  at  the  value  i  =  -  ,  that  is,  in  the  first  moment  after 

closing  the  circuit  all  the  impressed  e.m.f.  is  consumed  by  the 
current  in  the  resistance,  since  no  charge  and  therefore  no 
potential  difference  exists  at  the  condenser.  With  increasing 
charge  of  the  condenser,  and  therefore  increasing  potential 
difference  at  the  condenser  terminals,  less  and  less  e.m.f.  is 
available  for  the  resistance,  and  the  current  decreases,  and 
ultimately  becomes  zero,  when  the  condenser  is  fully  charged. 

If  the  circuit  also  contains  inductance  L,  then  the  current 
cannot  rise  instantly  but  only  gradually:  in  the  moment  after 
closing  the  circuit  the  potential  difference  at  the  condenser  is 
still  zero,  and  rises  at  such  a  rate  that  the  increase  of  magnetic 
flux  iL  in  the  inductance  produces  an  e.m.f.  Ldi/dt,  which 
consumes  the  impressed  e.m.f.  Gradually  the  potential  differ- 
ence at  the  condenser  increases  with  its  increasing  charge,  and 
the  current  and  thereby  the  e.m.f.  consumed  by  the  resistance 
increases,  and  so  less  e.m.f.  being  available  for  consumption  by 
the  inductance,  the  current  increases  more  slowly,  until  ulti- 
mately it  ceases  to  rise,  has  reached  a  maximum,  the  inductance 
consumes  no  e.m.f.,  but  all  the  impressed  e.m.f.  is  consumed  by 
the  current  in  the  resistance  and  by  the  potential  difference  at 
the  condenser.  The  potential  difference  at  the  condenser  con- 
tinues to  rise  with  its  increasing  charge;  hence  less  e.m.f.  is 
available  for  the  resistance,  that  is,  the  current  decreases  again, 
and  ultimately  becomes  zero,  when  the  condenser  is  fully 
charged.  During  the  decrease  of  current  the  decreasing  mag- 
netic flux  iL  in  the  inductance  produces  an  e.m.f.,  which  assists 
the  impressed  e.m.f.,  and  so  retards  somewhat  the  decrease  of 
current. 

Fig.  2  shows  the  charging  current  of  a  condenser  through  an 
inductive  circuit,  as  i,  and  the  potential  difference  at  the  con- 
denser terminals,  as  e,  with  a  continuous  impressed  e.m.f.  e0, 
for  the  circuit  constants  r  =  250  ohms;  L  =  100  mh.;  C  = 
10  mf.,  and  e0  =  1000  volts. 

If  the  resistance  is  very  small,  the  current  immediately  after 


INTRODUCTION 


19 


closing  the  circuit  rises  very  rapidly,  quickly  charges  the  con- 
denser, but  at  the  moment  where  the  condenser  is  fully  charged 
to  the  impressed  e.m.f.  e0,  current  still  exists.  This  current 
cannot  instantly  stop,  since  the  decrease  of  current  and  there- 
with the  decrease  of  its  magnetic  flux  iL  generates  an  e.m.f., 


1000 


4 — 800 


Fig.  2.     Charging  a  condenser  through  a  circuit  having  resistance  and 
inductance.    Constant  potential.    Logarithmic  charge:  high  resistance. 

which  maintains  the  current,  or  retards  its  decrease.  Hence 
electricity  still  continues  to  flow  into  the  condenser  for  some 
time  after  it  is  fully  charged,  and  when  the  current  ultimately 
stops,  the  condenser  is  overcharged,  that  is,  the  potential  dif- 
ference at  the  condenser  terminals  is  higher  than  the  impressed 
e.m.f.  e0,  and  as  result  the  condenser  has  partly  to  discharge 
again,  that  is,  electricity  begins  to  flow  in  the  opposite  direction, 
or  out  of  the  condenser.  In  the  same  manner  this  reverse 
current,  due  to  the  inductance  of  the  circuit,  overreaches  and 
discharges  the  condenser  farther  than  down  to  the  impressed 
e.m.f.  e0,  so  that  after  the  discharge  current  stops  again  a  charg- 
ing current  —  now  less  than  the  initial  charging  current - 
starts,  and  so  by  a  series  of  oscillations,  overcharges  and  under- 
charges, the  condenser  gradually  charges  itself,  and  ultimately 
the  current  dies  out. 

Fig.  3  shows  the  oscillating  charge  of  a  condenser  through  an 
inductive  circuit,  by  a  continuous  impressed  e.m.f.  e0.  The 
current  is  represented  by  i,  the  potential  difference  at  the  con- 
denser terminals  by  e,  with  the  time  as  abscissas.  The  con- 
stants of  the  circuit  are:  r  =  40  ohms;  L  =  100  mh.;  C  = 
10  mf.,  and  eQ  =  1000  volts. 

In  such  a  continuous-current  circuit,  containing  resistance, 
inductance,  and  capacity  in  series  to  each  other,  the  current  at 
the  moment  of  closing  the  circuit  as  well  as  the  final  current 


20 


TRANSIENT  PHENOMENA 


is  zero,  but  a  current  exists  immediately  after  closing  the 
circuit,  as  a  transient  phenomenon;  a  temporary  current, 
steadily  increasing  and  then  decreasing  again  to  zero,  or  con- 
sisting of  a  number  of  alternations  of  successively  decreasing 
amplitude :  an  oscillating  current. 

If  the  circuit  contains  no  resistance  and  inductance,  the  cur- 
rent into  the  condenser  would  theoretically  be  infinite.     That 


f 

s 

N 

6       1200 

/ 

A 

\ 

\ 

^ 

x^ 

*        H 

4    7?  8004—  f 

1 

\ 

\ 

^\? 

-  — 

4     oSOOJ-4 

/ 

\ 

^ 

x* 

!^° 

«• 

=1000 

vo 
oh 
ml 

ts 

o    ,     Ann  1  1 

L4=  100 

, 

a            lJyJ7 

\ 

. 

V 

\ 

c 

= 

10 

mf. 

\6 

=, 

Dt 

gr^es 

AT 

S 

k 

r^ 

•—  ^ 

8 

u 

It 

U\ 
' 

"2A 

tO 

8 

!7 

4 

0 

45 

0 

5( 

< 

&J 

^s 

^2 

0 

/ 

\ 

1 

-4       .        1  — 

\ 

s  t 

/ 

Fig.  3.     Charging  a  condenser  through  a  circuit  having  resistance  and 
inductance.     Constant  potential.     Oscillating  charge:  low  resistance. 

is,  with  low  resistance  and  low  inductance,  the  charging  current 
of  a  condenser  may  be.  enormous,  and  therefore,  although  only 
transient,  requires  very  serious  consideration  and  investigation. 
If  the  resistance  is  very  low  and  the  inductance  appreciable, 
the  overcharge  of  the  condenser  may  raise  its  voltage  above 
the  impressed  e.m.f.,e0  sufficiently  to  cause  disruptive  effects. 
14.  If  an  alternating  e.m.f., 

e  =  E  cos  0, 

is  impressed  upon  a  circuit  of  such  constants  that  the  current 
lags  45°,  that  is,  the  current  is 

i  =  I  cos  (d  -  45°), 

and  the  circuit  is  closed  at  the  moment  6  =  45°,  at  this 
moment  the  current  should  be  at  its  maximum  value.  It  is, 
however,  zero,  and  since  in  a  circuit  containing  inductance  (that 
is,  in  practically  any  circuit)  the  current  cannot  change  instantly, 
it  follows  that  in  this  case  the  current  gradually  rises  from  zero 
as  initial  value  to  the  permanent  value  of  the  sine  wave  i. 
This  approach  of  the  current  from  the  initial  value,  in  the 


INTRODUCTION 


21 


present  case  zero,  to  the  final  value  of  the  curve  i,  can  either 
be  gradual,  as  shown  by  the  curve  il  of  Fig.  4,  or  by  a  series 
of  oscillations  of  gradually  decreasing  amplitude,  as  shown  by 
curve  i2  of  Fig.  4. 

15.   The  general  solution  of  an  electric  current  problem  there- 
fore includes  besides  the  permanent  term,  constant  or  periodic, 


l 


/i 


>c- 


Gradual  or  Logarithm  o  start  of  current: 


Oscillatory  or  1 


S 


arjthui  e  start 
rigonometrio  s 

I**™** 


[rtartV 


Fig.  4.     Starting  of  an  alternating-current  circuit  having  inductance. 

a  transient  term,  which  disappears  after  a  time  depending  upon 
the  circuit  conditions,  from  an  extremely  small  fraction  of  a 
second  to  a  number  of  seconds. 

These  transient  terms  appear  in  closing  the  circuit,  opening 
the  circuit,  or  in  any  other  way  changing  the  circuit  conditions, 
as  by  a  change  of  load,  a  change  of  impedance,  etc. 

In  general,  in  a  circuit  containing  resistance  and  inductance 
only,  but  no  capacity,  the  transient  terms  of  current  and  volt- 
age are  not  sufficiently  large  and  of  long  duration  to  cause 
harmful  nor  even  appreciable  effects,  and  it  is  mainly  in  circuits 
containing  capacity  that  excessive  values  of  current  and  poten- 
tial difference  may  be  reached  by  the  transient  term,  and  there- 
with serious  results  occur.  The  investigation  of  transient  terms 
therefore  is  largely  an  investigation  of  the  effects  of  electro- 
static capacity. 

16.  No  transient  terms  result  from  the  resistance,  but  only 
those  circuit  constants  which  represent  storage  of  energy,  mag- 
netically by  the  inductance  L,  electrostatically  by  the  capacity 
C,  give  rise  to  transient  phenomena,  and  the  more  the  resist- 


22  TRANSIENT  PHENOMENA 

ance  predominates,  the  less  is  therefore  the  severity  and  dura- 
tion of  the  transient  term. 

When  closing  a  circuit  containing  inductance  or  capacity 
or  both,  the  energy  stored  in  the  inductance  and  the  capacity 
has  first  to  be  supplied  by  the  impressed  e.m.f.  before  the 
circuit  conditions  can  become  stationary.  That  is,  in  the  first 
moment  after  closing  an  electric  circuit,  or  in  general  changing 
the  circuit  conditions,  tne  impressed  e.m.f.,  or  rather  the  source 
producing  the  impressed  e.m.f.,  has,  in  addition  to  the  power 
consumed  in  maintaining  the  circuit,  to  supply  the  power  which 
stores  energy  in  inductance  and  capacity,  and  so  a  transient 
term  appears  immediately  after  any  change  of  circuit  condi- 
tion. If  the  circuit  contains  only  one  energy-storing  constant, 
as  either  inductance  or  capacity,  the  transient  term,  which 
connects  the  initial  with  the  stationary  condition  of  the  circuit, 
necessarily  can  be  a  steady  logarithmic  term  only,  or  a  gradual 
approach.  An  oscillation  can  occur  only  with  the  existence  of 
two  energy-storing  constants,  as  capacity  and  inductance,  which 
permit  a  surge  of  energy  from  the  one  to  the  other,  and  there- 
with an  overreaching. 

17.  Transient  terms  may  occur  periodically  and  in  rapid  suc- 
cession, as  when  rectifying  an  alternating  current  by  synchro- 
nously reversing  the  connections  of  the  alternating  impressed 
e.m.f.  with  the  receiver  circuit  (as  can  be  done  mechanically 
or  without  moving  apparatus  by  undirectional  conductors,  as 
arcs).    At  every  half  wave  the  circuit  reversal  starts  a  tran- 
sient term,  and  usually  this  transient  term  has  not  yet  disap- 
peared, frequently  not  even  greatly  decreased,  when  the  next 
reversal  again  starts  a  transient  term.     These  transient  terms 
may  predominate  to  such  an  extent  that  the  current  essentially 
consists  of  a  series  of  successive  transient  terms. 

18.  If  a  condenser  is  charged  through  an  inductance,  and  the 
condenser  shunted  by  a  spark  gap  set  for  a  lower  voltage  than 
the  impressed,  then  the  spark  gap  discharges  as  soon  as  the 
condenser  charge  has  reached  a  certain  value,  and  so  starts  a 
transient  term;  the  condenser  charges  again,  and  discharges, 
and  so  by  the  successive  charges  and  discharges  of  the  condenser 
a  series  of  transient  terms  is  produced,  recurring  at  a  frequency 
depending  upon  the  circuit  constants  and  upon  the  ratio  of  the 
disruptive  voltage  of  the  spark  gap  to  the  impressed  e.m.f. 


INTRODUCTION 


23 


>uch  a  phenomenon  for  instance  occurs  when  on  a  high- 
potential  alternating-current  system  a  weak  spot  appears  in  the 
cable  insulation  and  permits  a  spark  discharge  to  pass  to  the 
ground,  that  is,  in  shunt  to  the  condenser  formed  by  the  cable 
conductor  and  the  cable  armor  or  ground. 

19.  In  most  cases  the  transient  phenomena  occurring  in 
electric  circuits  immediately  after  a  change  of  circuit  conditions 
are  of  no  importance,  due  to  their  short  duration.  They 
require  serious  consideration,  however,  - 

(a)  In  those  cases  where  they  reach  excessive  values.  Thus 
in  connecting  a  large  transformer  to  an  alternator  the  large 
initial  value  of  current  may  do  damage.  In  short-circuiting  a 
large  alternator,  while  the  permanent  or  stationary  short-circuit 
current  is  not  excessive  and  represents  little  power,  the  very 
much  larger  momentary  short-circuit  current  may  be  beyond 
the  capacity  of  automatic  circuit-opening  devices  and  cause 
damage  by  its  high  power.  In  high-potential  transmissions 
the  potential  differences  produced  by  these  transient  terms  may 
reach  values  so  high  above  the  normal  voltage  as  to  cause 
disruptive  effects. 

(6)  Lightning,  high-potential  surges,  etc.,  are  in  their  nature 
essentially  transient  phenomena,  usually  of  oscillating  character. 

(c)  The  periodical  production  of  transient  terms  of  oscillating 
character  is  one  of  the  foremost  means  of  generating  electric  cur- 
rents of  very  high  frequency  as  used  in  wireless  telegraphy,  etc. 

(d)  In  alternating-current  rectifying  apparatus,  by  which  the 
direction  of  current  in  a  part  of  the  circuit  is  reversed  every  half 
wave,  and  the  current  so  made  unidirectional,  the  stationary 
condition  of  the  current  in  the  alternating  part  of  the  circuit  is 
usually  never  reached,  and  the  transient  term  is  frequently  of 
primary  importance. 

(e)  In   telegraphy   the   current  in  the   receiving  apparatus 
essentially  depends  on  the  transient  terms,  and  in  long-distance 
cable  telegraphy  the  stationary  condition  of  current  is  never 
approached,  and  the  speed  of  telegraphy  depends  on  the  duration 
of  the  transient  terms. 

(f)  Phenomena  of  the  same  character,  but  with  space  instead 
of  time  as  independent  variable,  are  the  distribution  of  voltage 
and  current  in  a  long-distance  transmission  line;  the  phenomena 
occurring  in  niultigap  lightning  arresters;  the  transmission  of 


24  TRANSIENT  PHENOMENA 

current  impulses  in  telephony;  the  distribution  of  alternating 
current  in  a  conductor,  as  the  rail  return  of  a  single-phase  railway; 
the  distribution  of  alternating  magnetic  flux  in  solid  magnetic 
material,  etc. 

Some  of  the  simpler  forms  of  transient  terms  are  investigated 
and  discussed  in  the  following  pages. 


CHAPTER  III. 

INDUCTANCE   AND    RESISTANCE   IN   CONTINUOUS- 
CURRENT   CIRCUITS. 

20.  In  continuous-current  circuits  the  inductance  does  not 
•enter  the  equations  of  stationary  condition,  but,  if  e0  =  impressed 
e.m.f.,  r  =  resistance,  L  =  inductance,  the  permanent  value  of 

/> 

current  is  ia  =  —  • 
r 

Therefore  less  care  is  taken  in  direct-current  circuits  to  reduce 
the  inductance  than  in  alternating-current  circuits,  where  the 
inductance  usually  causes  a  drop  of  voltage,  and  direct-current 
circuits  as  a  rule  have  higher  inductance,  especially  if  the  circuit 
is  used  for  producing  magnetic  flux,  as  in  solenoids,  electro- 
magnets, machine-fields. 

Any  change  of  the  condition  of  a  continuous-current  circuit, 
as  a  change  of  e.m.f.,  of  resistance,  etc.,  which  leads  to  a  change 
of  current  from  one  value  i0  to  another  value  iv  results  in  the 
appearance  of  a  transient  term  connecting  the  current  values 
i0  and  iv  and  into  the  equation  of  the  transient  term  enters  the 
inductance. 

Count  the  time  t  from  the  moment  when  the  change  in  the 
continuous-current  circuit  starts,  and  denote  the  impressed 
e.m.f.  by  e0,  the  resistance  by  r,  and  the  inductance  by  L. 

p 

il  =  -  =  current  in  permanent  or  stationary  condition  after 

the  change  of  circuit  condition. 

Denoting  by  i0  the  current  in  circuit  before  the  change,  and 
therefore  at  the  moment  t  =  0,  by  i  the  current  during  the 
change,  the  e.m.f.  consumed  by  resistance  r  is 

ir, 
and  the  e.m.f.  consumed  by  inductance  L  is 

di 

Ldt' 
where  i  =  current  in  the  circuit. 

26 


26  TRANSIENT  PHENOMENA 

di 

Hence,  eQ  =  ir  +  L  —  >  (1) 

dt 

or,  substituting  eQ  =  if,  and  transposing, 

-i*-i±V 

This  equation  is  integrated  by 

-  -t  =  log  (i  -  ij  -  logc, 
where  —  log  c  is  the  integration  constant,  or, 

r 

i  —  i^  =  ce    L  . 
However,  for  t  =  0,  i  =  iQ. 
Substituting  this,  gives 

IQ  —  il  =  c, 

-ft 
hence,  i  =  il  +  (i0  -  t\)  e      '   ,  (3) 

the  equation  of  current  in  the  circuit. 
The  counter  e.m.f.  of  self -inductance  is 

e^-L^rtf.-*,).^',  .          (4) 

hence  a  maximum  for  t  =  0,  thus : 

«i°  =  r  (i,  -  i,).  (5) 

The  e.m.f.  of  self-inductance  ex  is  proportional  to  the  change 
of  current  (i0  —  t\),  and  to  the  resistance  r  of  the  circuit  after 
the  change,  hence  would  be  <*>  for  r  =  <*> ,  or  when  opening  the 
circuit.  That  is,  an  inductive  circuit  cannot  be  opened  instantly, 
but  the  arc  following  the  break  maintains  the  circuit  for  some 
time,  and  the  voltage  generated  in  opening  an  inductive  circuit 
is  the  higher  the  quicker  the  break.  Hence  in  a  highly  inductive 
circuit,  as  an  electromagnet  or  a  machine  field,  the  insulation 
may  be  punctured  by  excessive  generated  e.m.f.  when  quickly 
opening  the  circuit. 

As  example,  some  typical  circuits  may  be  considered. 


CONTINUOUS-CURRENT  CIRCUITS  27 

21.  Starting  of  a  continuous-current  lighting  circuit,  or  non-in- 
ductive load. 

Let  e0  =  125  volts  =  impressed  e.m.f.  of  the  circuit,  and 
tj  «  1000  amperes  =  current  in  the  circuit  under  stationary 
condition;  then  the  effective  resistance  of  the  circuit  is 

=  0.125  ohm. 

Assuming  10  per  cent  drop  in  feeders  and  mains,  or  12.5  volts, 
gives  a  resistance,  r0  =  0.0125  ohm  of  the  supply  conductors. 
In  such  large  conductor  the  inductance  may  be  estimated  as 
10  mh.  per  ohm;  hence,  L  =  0.125  mh.  =  0.000125  henry. 

The  current  at  the  moment  of  starting  is  i0  =  0,  and  the  general 
equation  of  the  current  in  the  circuit  therefore  is,  by  substitution 

m  (3))  i  =  1000  (1  -  e-1000')-  (6) 

The  time  during  which  this  current  reaches  half  value,  or 
i  =  500  amperes,  is  given  by  substitution  in  (6) 

500  =  1000  (1  -  s-1"*"), 
hence  e"1000'  =  0.5, 

t  =  0.00069  seconds. 

The  time  during  which  the  current  reaches  90  per  cent  of  its 
full  value,  or  i  =  900  amperes,  is  t  =  0.0023  seconds,  that  is, 
the  current  is  established  in  the  circuit  in  a  practically  inappre- 
ciable time,  a  fraction  of  a  hundredth  of  a  second. 

22.  Excitation  of  a  motor  field. 

Let,  in  a  continuous-current  shunt  motor,  e0  =  250  volts  = 
impressed  e.m.f.,  and  the  number  of  poles  =  8. 

Assume  the  magnetic  flux  per  pole,  <l>0  =  12.5  megalines,  and 
the  ampere-turns  per  pole  required  to  produce  this  magnetic 
flux  as  $  =  9000. 

Assume  1000  watts  used  for  the  excitation  of  the  motor 
field  gives  an  exciting  current 

1000 
h  =-          =  4  amperes, 


and  herefrom  the  resistance  of  the  total  motor  field  circuit  is 
r  =  e-?  =  62.5  ohms. 


28  TRANSIENT  PHENOMENA 

To  produce  JF  =  9000  ampere-turns,  with   il  =  4  amperes, 

cjr 

requires  —  =  2250  turns  per  field  spool,  or  a  total  of  n  =  18,000 

l\ 
turns. 

n  =  18,000  turns  interlinked  with  <I>0  =  12.5  megalines  gives 
a  total  number  of  interlinkages  for  i1  =  4  amperes  of  n<&0  = 
225  X  109,  or  562.5  X  109  interlinkages  per  unit  current,  or 
10  amperes,  that  is,  an  inductance  of  the  motor  field  circuit 
L  =  562.5  henrys. 

The  constants  of  the  circuit  thus  are  e0  =  250  volts;  r  =  62.5 
ohms;  L  =  562.5  henrys,  and  i0  =  0  =  current  at  time  t  =  0. 

Hence,  substituting  in  (3)  gives  the  equation  of  the  exciting 
current  of  the  motor  field  as 

'      i  =  4  (1  _  e-o-m")  (7) 

Half  excitation  of  the  field  is  reached  after  the  time  t  =  6.23 
seconds; 

90  per  cent  of  full  excitation,  or  i  =  3.6  amperes,  after  the 
time  t  =  20.8  seconds. 

That  is,  such  a  motor  field  takes  a  very  appreciable  time 
after  closing  the  circuit  before  it  has  reached  approximately 
full  value  and  the  armature  circuit  may  safely  be  closed. 

Assume  now  the  motor  field  redesigned,  or  reconnected  so 
as  to  consume  only  a  part,  for  instance  half,  of  the  impressed 
e.m.f.,  the  rest  being  consumed  in  non-inductive  resistance. 
This  may  be  done  by  connecting  the  field  spools  by  two  in 
multiple. 

In  this  case  the  resistance  and  the  inductance  of  the  motor 
field  are  reduced  to  one-quarter,  but  the  same  amount  of 
external  resistance  has  to  be  added  to  consume  the  impressed 
e.m.f.,  and  the  constants  of  the  circuit  then  are:  e0  =  250 
volts;  r  =  31.25  ohms;  L  =  140.6  henrys,  and  i0  =  0. 

The  equation  of  the  exciting  current  (3)  then  is 

i  =  8  (1  -  e"  °'22220,  (8) 

that  is,  the  current  rises  far  more  rapidly.     It  reaches  0.5 
value  after  t  =  3.11  seconds,  0.9  value  after  t  =  10.4  seconds. 

An  inductive  circuit,  as  a  motor  field  circuit,  may  be  made 
to  respond  to  circuit  changes  more  rapidly  by. inserting  non- 
inductive  resistance  in  series  with  it  and  increasing  the  im- 


CONTINUOUS-CURRENT  CIRCUITS  29 

pressed  e.m.f.,  that  is,  the  larger  the  part  of  the  impressed 
e.m.f.  consumed  by  non-inductive  resistance,  the  quicker  is  the 
change. 

Disconnecting  the  motor  field  winding  from  the  impressed 
e.m.f.  and  short-circuiting  it  upon  itself,  as  by  leaving  it  con- 
nected in  shunt  with  the  armature  (the  armature  winding 
resistance  and  inductance  being  negligible  compared  with  that 
of  the  field  winding),  causes  the  field  current  and  thereby  the 
field  magnetism  to  decrease  at  the  same  rate  as  it  increased  in 
(7)  and  (8),  provided  the  armature  instantly  comes  to  a  stand- 
still, that  is,  its  e.m.f.  of  rotation  disappears.  This,  however, 
is  usually  not  the  case,  but  the  motor  armature  slows  down 
gradually,  its  momentum  being  consumed  by  friction  and  other 
losses,  and  while  still  revolving  an  e.m.f.  of  gradually  decreas- 
ing intensity  is  generated  in  the  armature  winding;  this  e.m.f. 
is  impressed  upon  the  field. 

The  discharge  of  a  motor  field  winding  through  the  armature 
winding,  after  shutting  off  the  power,  therefore  leads  to  the 
case  of  an  inductive  circuit  with  a  varying  impressed  e.m.f. 

23.   Discharge  of  a  motor  field  winding. 

Assume  that  in  the  continuous-current  shunt  motor  dis- 
cussed under  22,  the  armature  comes  to  rest  tl  =  40  seconds 
after  the  energy  supply  has  been  shut  off  by  disconnecting  the 
motor  from  the  source  of  impressed  e.m.f.,  while  leaving  the 
motor  field  winding  still  in  shunt  with  the  motor  armature 
winding. 

The  resisting  torque,  which  brings  the  motor  to  rest,  may  be 
assumed  as  approximately  constant,  and  therefore  the  deceler- 
ation of  the  motor  armature  as  constant,  that  is,  the  motor 
speed  decreasing  proportionally  to  the  time. 

If  then  S  =  full  motor  speed,  S  (l  -  - j  is  the  speed  of  the 

motor  at  the  time  t  after  disconnecting  the  motor  from  the 
source  of  energy. 

Assume  the  magnetic  flux  3>  of  the  motor  as  approximately 
proportional  to  the  exciting  current,  at  exciting  current  i  the 

magnetic  flux  of  the  motor  is  <&=  ^<&0,  where  4>0=  12.5  mega- 
lines  is  the  flux  corresponding  to  full  excitation  it  =  4  amperes. 


30  TRANSIENT  PHENOMENA 

The  e.m.f.   generated  in  the  motor  armature  winding  and 
thereby  impressed  upon  the  field  winding  is  proportional  to 

the  magnetic  flux  of  the  field,  <I>,  and  to  the  speed  $  (l J, 

V 
and  since  full  speed  S  and  full  flux  <I>0  generate  an  e.m.f.  e0  = 

250  volts,  the  e.m.f.  generated  by  the  flux  <£  and  speed  S 1 1  -  -\> 
that  is,  at  time  t  is 


and  since 
we  have 


-«-r 

/     4 


e  =  ir  (l  -  -J;  (10) 

or  for  r  =  62.5  ohms,  and  tl  =  40  seconds,  we  have 

e  =  62.5^  (1  -  0.025  t).  (11) 

Substituting  this  equation  (10)  of  the  impressed  e.m.f.  into 
the  differential  equation  (1)  gives  the  equation  of  current  i 
during  the  field  discharge, 


henC6<  rtdt       di 


integrated  by 


where  the  integration  constant  c  is  found  by 
hence, 


t  =  0,    i  =  iv    log  cii  =  0,     c  =  -  , 


2 


(15) 


CONTINUOUS-CURRENT  CIRCUITS  31 

This  is  the  equation  of  the  field  current  during  the  time  in 
which  the  motor  armature  gradually  comes  to  rest. 
At  the  moment  when  the  motor  armature  stops,  or  for 


it  is  rtl 

i2  =  v~^.  (16) 

This  is  the  same  value  which  the  current  would  have  with 
the  armature  permanently  at  rest,  that  is,  without  the  assistance 

of  the  e.m.f.  generated  by  rotation,  at  the  time  t  =  —  • 

The  rotation  of  the  motor  armature  therefore  reduces  the 
decrease  of  field  current  so  as  to  require  twice  the  time  to  reach 
value  i2,  that  it  would  without  rotation. 

These  equations  cease  to  apply  for  t  >  tv  that  is,  after  the 
armature  has  come  to  rest,  since  they  are  based  on  the  speed 

equation  S  ( 1 J ,  and    this   equation   applies   only  up    to 

t  =  tv  but  for  t  >  tj_  the  speed  is  zero,  and  not  negative,  as 
given  by  S  ( 1  -  -j  • 

That  is,  at  the  moment  t  =  ^  a  break  occurs  in  the  field 
discharge  curve,  and  after  this  time  the  current  i  decreases  in 
accordance  with  equation  (3),  that  is, 

L  \         )  /-i  >7\ 

I   =  12£  >  (17) 

or,  substituting  (16), 

i  =  v"1 '*'*'.  (18) 

Substituting  numerical  values  in  these  equations  gives : 
for  t  <  tv 

i=   4  £-0.001388*'.  (19) 

for  t  =  tl  =  40, 

i  =  0.436;  (20) 

for  t  >  tv 

i  =  4  r  °-mi  (<-2o).  (21) 


32 


TRANSIENT  PHENOMENA 


Hence,  the  field  has  decreased  to  half  its  initial  value  after 
the  time  t  =  22.15  seconds,  and  to  one  tenth  of  its  initial 
value  after  t  =  40.73  seconds. 


40  seconds 


5        10       15        20 

Seconds 

Fig.  5.     Field  discharge  current. 

Fig.  5  shows  as  curve  I  the  field  discharge  current,  by  equations 
(19),  (20),  (21),  and  as  curve  II  the  current  calculated  by  the 
equation 

i  =  4£-<M111<, 

that  is,  the  discharge  of  the  field  with  the  armature  at  rest,  or 
when  short-circuited  upon  itself  and  so  not  assisted  by  the 
e.m.f.  of  rotation  of  the  armature. 

The  same  Fig.  5  shows  as  curve  III  the  beginning  of  the  field 
discharge  current  for  L  =  4200,  that  is,  the  case  that  the  field 
circuit  has  a  much  higher  inductance,  as  given  by  the  equation 

i    __    4  £—  0-000185 1- 

As  seen  in  the  last  case,  the  decrease  of  field  current  is  very  slow, 
the  field  decreasing  to  half  value  in  47.5  seconds. 

24.  Self-excitation  of  direct-current  generator. 

In  the  preceding,  the  inductance  L  of  the  machine  has  been 
assumed  as  constant,  that  is,  the  magnetic  flux  <£  as  proportional 
to  the  exciting  current  i.  For  higher  values  of  <£,  this  is  not 
even  approximately  the  case.  The  self-excitation  of  the  direct- 
current  generator,  shunt  or  series  wound,  that  is,  the  feature 


CONTINUOUS-CURRENT  CIRCUITS  33 

that  the  voltage  of  the  machine  after  the  start  gradually  builds 
up  from  the  value  given  by  the  residual  magnetism  to  its  full 
value,  depends  upon  the  disproportionality  of  the  magnetic  flux 
with  the  magnetizing  current.  When  considering  this  phenom- 
enon, the  inductance  cannot  therefore  be  assumed  as  constant. 

When  investigating  circuits  in  which  the  inductance  L  is  not 
constant  but  varies  with  the  current,  it  is  preferable  not  to  use 
the  term  "  inductance"  at  all,  but  to  introduce  the  magnetic 
flux  4>. 

The  magnetic  flux  ®  varies  with  the  magnetizing  current  i  by 
an  empirical  curve,  the  magnetic  characteristic  or  saturation 
curve  of  the  machine.  This  can  approximately,  within  the  range 
considered  here,  be  represented  by  a  hyperbolic  curve,  as  was 
first  shown  by  Frohlich  in  1882  : 

*--'  (22) 


where  <f>  =  magnetic  flux  per  ampere,  in  megalines,  at  low 
density. 

—  =  magnetic  saturation  value,  or  maximum  magnetic  flux, 
in  megalines,  and 

£.!+«  (23) 

can  be  considered  as  the  magnetic  exciting  reluctance  of  the 
machine  field  circuit,  which  here  appears  as  linear  function  of 
the  exciting  current  i. 

Considering  the  same  shunt-wound  commutating  machine  as 
in  (12)  and  (13),  having  the  constants  r  =  62.5  ohms  =  field 
resistance;  <l>0  =  12.5  megalines  =  magnetic  flux  per  pole  at 
normal  m.m.f.;  SF  =  9000  ampere-turns  =  normal  m.m.f.  per 
pole;  n  =  18,000  turns  =  total  field  turns  (field  turns  per  pole 

=  18'QQQ  =  2250),    and    ^  =  4    amperes  =  current    for    full 

8 
excitation,  or  flux,  4>0  =  12.5  megalines. 

Assuming  that  at  full  excitation,  4>0,  the  magnetic  reluctance 
has  already  increased  by  50  per  cent  above  its  initial  value,  that 


34  TRANSIENT  PHENOMENA 

ampere-turns          i 

is.  that  the  ratio  -  .    _ — >  or  — ,  at  <P  =  4>   =  12.5  mega- 

magnetic  flux          <J> 

lines  and  i  =  il  =  4  amperes,  is  50  per  cent  higher  than  at  low 
excitation,  it  follows  that 

1  +  bi,  =  1.5, 


or 


b  =  0.125. 


(24) 


Since  i  =  iv  =  4  produces  $  =  $0  =  12.5,  it   follows,  from 
(22)  and  (24) 

<£  -  4.69. 

That  is,  the  magnetic  characteristic  (22)  of  the  machine  is 
approximated  by 

- 


Let  now  ec  =  e.m.f.  generated  by  the  rotation  of  the  arma- 
ture per  megaline  of  field  flux. 

This  e.m.f.  ec  is  proportional  to  the  speed,  and  depends  upon 
the  constants  of  the  machine.  At  the  speed  assumed  in  (12) 
and  (13),  $0  =  12.5  megalines,  e0  =  250  volts,  that  is, 

ec  =  -^  =  20  volts. 
$0 

Then,  in  the  field  circuit  of  the  machine,  the  impressed  e.m.f., 
or  e.m.f.  generated  in  the  armature  by  its  rotation  through  the 
magnetic  field  is, 

e  =  ec<b  =  20$; 

the  e.m.f.  consumed  by  the  field  resistance  r  is 

ir  =  62.5  i; 

the  e.m.f.  consumed  by  the  field  inductance,  that  is,  generated 
in  the  field  coils  by  the  rise  of  magnetic  flux  <£,  is 


(<i>  being  given  in  megalines,  eQ  in  volts.) 


CONTINUOUS-CURRENT  CIRCUITS  35 

The  differential  equation  of  the  field  circuit  therefore  is  (1) 

(26) 


n 


Since  this  equation  contains  the  differential  quotient  of  4>,  it 
is  more  convenient  to  make  4>  and  not  i  the  dependent  variable; 
then  substitute  for  i  from  equation  (22), 


which  gives 


or,  transposed, 


100  ~dt  ' 


100 
n 


(27) 
(28) 
(29) 


This  equation  is  integrated  by  resolving  into  partial  fraction 
by  the  identity 


ec-  r)  -  bec 
resolved,  this  gives 

hence,  A 


B 


<j>ec  -  r  -  bec 


-r)-  (Abec  *  -  B  *); 


(30) 


<t*c  ~  r 
br 


and 


100 


+ 


.  —  r)  (<l>ec  —  r  —  be, 
This  integrates  by  the  logarithmic  functions 
100 1          4  r 


(31) 


(32) 


n 


ec(<t>ec  -  r} 


log(<j>ec-r-bec$>)+C.    (33) 


36  TRANSIENT  PHENOMENA 

The  integration  constant  C  is  calculated  from  the  residual 
magnetic  flux  of  the  machine,  that  is,  the  remanent  magnetism 
of  the  field  poles  at  the  moment  of  start. 

Assume,  at  the  time,  t  =  0,  4>  =  4>r  =  0.5  megalines  =  residual 
magnetism  and  substituting  in  (33), 


and  herefrom  calculate  C. 
C  substituted  in  (33)  gives 

100  1          <}>  <$>  r  <l>ec-r-  bec3> 


n 
or, 


substituting 
and 


n 


em  = 


where  em  =  e.m.f.  generated  in  the  armature  by  the  rotation  in 
the  residual  magnetic  field, 

n  (  e  (bec  —  r  —  be 

(36) 


This,  then,  is  the  relation  between  e  and  t,  or  the  equation 
of  the  building  up  of  a  continuous-current  generator  from  its 
residual  magnetism,  its  speed  being  constant. 

Substituting  the  numerical  values  n  =  18,000  turns;  (f>  = 
4.69  megalines;  b  =  0.125;  ec  =  20  volts;  r  =  62.5  ohms;  4>r  = 
0.5  megaline,  and  em  =  10  volts,  we  have 

t  =  26.8  log  $  -  17.9  log  (31.25  -  2.5  $)  +  79.6      (37) 

and 

t  =  26.8  log  e  -  17.9  log  (31.25  -  0.125  e)  -  0.8.      (38) 


CONTINUOUS-CURRENT  CIRCUITS 


37 


Fig.  6  shows  the  e.m.f.  e  as  function  of  the  time  t.  As  seen, 
under  the  conditions  assumed  here,  it  takes  several  minutes 
before  the  e.m.f.  of  the  machine  builds  up  to  approximately 
full  value. 


0   20   40   60   80   100  120  140  760  ISO  200  Sec. 

Fig.  6.     Building-up  curve  of  a  shunt  generator. 

The  phenomenon  of  self-excitation  of  shunt  generators  there- 
fore is  a  transient  phenomenon  which  may  be  of  very  long 
duration. 

From  equations  (35)  and  (36)  it  follows  that 

<l>ec  -  r 


e  = 


250  volts 


(39) 


is  the  e.m.f.  to  which  the  machine  builds  up  at  t  =  o>,  that  is, 
in  stationary  condition. 
To  make  the  machine  self-exciting,  the  condition 


fa  -  r  >  0 
must  obtain,  that  is,  the  field  winding  resistance  must  be 

r  <  fa 
or, 


(40) 


(41) 


r  <  93.8  ohms, 
or,  inversely,  ec,  which  is  proportional  to  the  speed,  must  be 

r 
*<>$' 

or, 

er>  13.3  volts. 


(42) 


38  TRANSIENT  PHENOMENA 

The  time  required  by  the  machine  to  build  up  decreases  with 
increasing  ec,  that  is,  increasing  speed;  and  increases  with 
increasing  r,  that  is,  increasing  field  resistance. 

25.   Self-excitation  of  direct-current  series  machine. 

Of  interest  is  the  phenomenon  of  self-excitation  in  a  series 
machine,  as  a  railway  motor,  since  when  using  the  railway  motor 
as  brake,  by  closing  its  circuit  upon  a  resistance,  its  usefulness 
depends  upon  the  rapidity  of  building  up  as  generator. 

Assuming  a  4-polar  railway  motor,  designed  for  e0  =  600  volts 
and  iv=  200  amperes,  let,  at  current  i  =  i1=  200  amperes,  the 
magnetic  flux  per  pole  of  the  motor  be  4>0  =  10  megalines,  and 
8000  ampere-turns  per  field  pole  be  required  to  produce  this 
flux.  This  gives  40  exciting  turns  per  pole,  or  a  total  of  n  = 
160  turns. 

Estimating  8  per  cent  loss  in  the  conductors  of  field  and 
armature  at  200  amperes,  this  gives  a  resistance  of  the  motor 
circuit  r0=  0.24  ohms. 

To  limit  the  current  to  the  full  load  value  of  ^  =  200  amperes, 
with  the  machine  generating  eQ=  600  volts,  requires  a  total 
resistance  of  the  circuit,  internal  plus  external,  of 

r  =  3  ohms, 

or  an  external  resistance  of  2.76  ohms. 
600  volts  generated  by  10  megalines  gives 

ec=  60  volts  per  megaline  per  field  pole. 

Since  in  railway  motors  at  heavy  load  the  magnetic  flux  is 
carried  up  to  high  values  of  saturation,  at  i±  =  200  amperes  the 
magnetic  reluctance  of  the  motor  field  may  be  assumed  as  three 
times  the  value  which  it  has  at  low  density,  that  is,  in  equation 

<22>'  .  1  +  W,  -  3, 

°r'  6  -  0.01, 

and  since  for  i  =  200,  <b  =  10,  we  have  in  (22) 

<£  =  0.15, 

hence,  $  =  - — '  (43) 

represents  the  magnetic  characteristic  of  the  machine. 


CONTINUOUS-CURRENT  CIRCUITS  39 

Assuming  a   residual   magnetism  of  10  per   cent,  or  <I>r  = 
1  megaline,  hence  em  =  ec4>r=  60  volts,  and  substituting  in 
equation  (36)  gives  n  =  160  turns;   <f>  =  0.15  megaline;  b  = 
0.01;  ec=  60  volts;  r  =  3  ohms;  4>r=  1  megaline,  and  em  = 
60  volts, 

t  =  0.04  log  e  -  0.01333  log  (600  -  e)  -  0.08.         (44) 

This  gives  for  e  =  300,  or  0.5  excitation,  t  =  0.072  seconds; 
and  for  e  =  540,  or  0.9  excitation,  t  =  0.117  seconds;  that  is, 
such  a  motor  excites  itself  as  series  generator  practically  instantly, 
or  in  a  small  fraction  of  a  second. 

The  lowest  value  of  ec  at  which  self-excitation  still  takes  place 
is  given  by  equation  (42)  as 

ec  =  ^  =  20> 

that  is,  at  one-third  of  full  speed. 

If  this  series  motor,  with  field  and  armature  windings  connected 
in  generator  position, — that  is,  reverse  position, — short-circuits 
upon  itself, 

r  =  0.24  ohms, 

we  have 

t  =  0.0274  log  e  -  0.00073  log  (876  -  e)  -  0.1075,        (45) 
that  is,  self-excitation  is  practically  instantaneous : 

e  =  300  volts  is  reached  after  t  =  0.044  seconds. 

Since    for  e  =  300  volts,  the  current   i  =      =  1250  amperes, 

the  power  is  p  =  ei  =  375  kw.,  that  is,  a  series  motor  short- 
circuited  in  generator  position  instantly  stops. 
Short-circuited   upon  itself,  r  =  0.24,  this  series  motor  still 

builds  up  at  ec  =  -  =  1.6,  and  since  at  full  load  speed  ec=  60, 

9 

ec  =  1.6  is  2.67  per  cent  of  full  load  speed,  that  is,  the  motor 
acts  as  brake  down  to  2.67  per  cent  of  full  speed. 

It  must  be  considered,  however,  that  the  parabolic  equation 
(22)  is  only  an  approximation  of  the  magnetic  characteristic, 


40  TRANSIENT  PHENOMENA 

and  the  results  based  on  this  equation  therefore  are  approximate 
only. 

One  of  the  most  important  transient  phenomena  of  direct- 
current  circuits  is  the  reversal  of  current  in  the  armature  coil 
short-circuited  by  the  commutator  brush  in  the  commutating 
machine.  Regarding  this,  see  "  Theoretical  Elements  of  Elec- 
trical Engineering,"  Part  II,  Section  B. 


CHAPTER  IV. 

INDUCTANCE    AND    RESISTANCE    IN    ALTERNATING- 
CURRENT   CIRCUITS. 

•  26.  In  alternating-current  circuits,  the  inductance  L,  or,  as 
it  is  usually  employed,  the  reactance  x  =  2  nfL,  where  /  =  fre- 
quency, enters  the  expression  of  the  transient  as  well  as  the 
permanent  term. 

At  the  moment  0  =  0,  let  the  e.m.f.  e  =  E  cos  (0  —  00)  be 
impressed  upon  a  circuit  of  resistance  r  and  inductance  L,  thus 
inductive  reactance  x  =  2  xfL;  let  the  time  6  =  2  xft  be  counted 
from  the  moment  of  closing  the  circuit,  and  00  be  the  phase  of 
the  impressed  e.m.f.  at  this  moment. 

In  this  case  the  e.m.f.  consumed  by  the  resistance  =  ir, 
where  i  =  instantaneous  value  of  current. 

The  e.m.f.  consumed  by  the  inductance  L  is  proportional 

to  L  and  to  the  rate  of  change  of  the  current,  — ,  thus,  is  L  — , 

at  at 

or,  by  substituting  6  =  2  nft,  x  =  2  nfL,  the  e.m.f.  consumed 

by  inductance  is  x  —  • 
do 

Since  e  =  E  cos  (0  —  00)  =  impressed  e.m.f., 

di 
E  cos  (6  -  00)  =  ir  +  x  —  (1) 

is  the  differential  equation  of  the  problem. 
This  equation  is  integrated  by  the  function 

i  =  7  cos  (6  -  d)  +  A£~a°,  (2) 

where  e  =  basis  of  natural  logarithms  =  2.7183. 

Substituting  (2)  in  (1), 

E  cos  (6  -  00)  =  Ir  cos  (6  -  i)  +  Ars~a0  -  Ix  sin  (d-d)-  Aaxs'"', 
or,  rearranged: 

(E  cos  00  -  Ir  cos  §  -  Ix  sin  d)  cos  0  +  (E  sin  00  -  Ir  sin  8 

+  /x  cos  d)  sin  0  -  ^e~a"  (ax  -  r)  =  0. 
41 


42 


TRANSIENT  PHENOMENA 


Since  this  equation  must  be  fulfilled  for  any  value  of  6,  if  (2) 
is  the  integral  of  (1),  the  coefficients  of  cos  6,  sin  0,  £~a9  must 
vanish  separately. 

That  is, 

E  cos  00  —  Ir  cos  d  —  Ix  sin  d  =  0, 

E  sin  00  -  Ir  sin  d  +  Ix  cos  d  =  0, 


and 
Herefrom  it  follows  that 


ax  —  r  =  0. 


Substituting  in  (3), 


r 
a=  - 


tan  0X  = 


and 
where 


(3) 


(4) 


(5) 


and 

and  herefrom 

and 


z  =  W2  + 

lag  angle  and  z  =  impedance  of  circuit,  we  have 
E  cos  00  -  70  cos  (#  -0^=0 

E  sin  00  -  7z  sin  (d  -  0X)  =  0, 


(6) 


Thus,  by  substituting  (4)  and  (6)  in  (2),  the  integral  equation 
becomes 

E  --« 

i  =  -  cos  (0  -  00  -  0X)  +  As    x  ,  (7) 

where  A  is  still  indefinite,  and  is  determined  by  the  initial  con- 
ditions of  the  circuit,  as  follows : 

for  0  =  0,        i  =  0; 

hence,  substituting  in  (7), 

E 
0  =  -cos  (00  +  0J  +  A, 


ALTERNATING-CURRENT  CIRCUITS  43 

or, 

A  -_|cos  &  +  *!),  <(8) 

z 

and,  substituted  in  (7), 

i  =  -z  |  cos  (I?  -  00-  0J-  i~x°  cos  (00  +  OJ  j  (9) 

is  the  general  expression  of  the  current  in  the  circuit. 

If  at  the  starting  moment  0  =  0  the  current  is    not    zero 
but  =  iw  we  have,  substituted  in  (7), 


A  =  ^--(508(0 


i  =--    cos  (d  -  60  -  ^)-cos  (00  +  0,)-         e*      .    (10) 


27.    The  equation  of  current  (9)  contains  a  permanent  term 

E 

—  cos  (0  —  00  —  dj,  which  usually  is  the  only  term  considered, 

E    -~e 
and  a  transient  term  —  e    x  cos  (00  +  0t). 

z 

The  greater  the  resistance  r  and  smaller  the  reactance  x,  the 


more  rapidly  the  term   :-  e    ;c  cos  (00  -f  0t)  disappears. 

This  transient  term  is  a  maximum  if  the  circuit  is  closed  at 
the  moment    00  =  —  6V  that  is,   at  the  moment    when  the 

E 
permanent  value  of  current,  —  cos  (0  —  00  —  0t),  should  be  a 

maximum,  and  is  then 


The  transient  term  disappears  if  the  circuit  is  closed  at  the 
moment  00  =  90°  —  Ov  or  when  the  stationary  term  of  current 
passes  the  zero  value. 


44 


TRANSIENT  PHENOMENA 


As  example  is  shown,  in  Fig.  7,  the  starting  of  the  current 
under  the  conditions  of  maximum  transient  term,  or  6Q  =  —  dlt 

X 

in  a  circuit  of  the  following  constants:    —  =  0.1,  corresponding 
approximately  to  a  lighting  circuit,  where  the  permanent  value 


GO 


<feN 


so 


Degrees 


120 


Fig.  7.     Starting  current  of  an  inductive  circuit. 


X 


CM 

of  current  is  reached  in  a  small  fraction  of  a  half  wave;  —  =0.5, 
corresponding  to  the  starting  of  an  induction  motor  with  rheo- 

*M 

stat  in  the  secondary  circuit;  —  =  1.5,  corresponding  to  an 
unloaded  transformer,  or  to  the  starting  of  an  induction  motor 
with  short-cifcuited  secondary,  and  —  =  10,  corresponding  to  a 
reactive  coil. 


/ 

, 

( 

*  ^ 

\ 

/ 

'  \ 

. 

/ 

^-^ 

V 

/ 

\ 

/ 

f\ 

\ 

X 

r 

4 

// 

\\ 

// 

\ 

I 

^ 

It 

\\ 

De 

1 
grqcs 

/ 

\\ 

I 

\J 

0" 

V 

\ 

180 

7 

1 

360 

\ 

\ 

540 

/ 

P 

720 

9 

\ 

9W) 

It 

6 

08(J 

IMO 

\ 

\ 

/ 

i 

V\ 

2 

S 

/ 

f\ 

1 

\ 

^^ 

n 

S 

^ 

y 

V 

k    ^ 

7 

J 

\ 

\ 

^ 

s>^ 

\ 

1 

1 

Fig.  8.     Starting  current  of  an  inductive  circuit. 

X 

Of  the  last  case,  —  =  10,  a  series  of  successive  waves  are 
r 

plotted  in  Fig.  8,  showing  the  very  gradual  approach  to  perma- 
nent condition. 


ALTERNATING-CURRENT  CIRCUITS 


45 


Fig.  9  shows,  for  the  circuit  —  =  1.5,  the  current  when  closing 

the  circuit  0°,  30°,  60°,  90°,  120°,  150°  respectively  behind  the 
zero  value  of  permanent  current. 

The  permanent  value  of  current  is  usually  shown  in  these 
diagrams  in  dotted  line. 


^^\Ss 


m 


1.5 


120         180         240          300 
Degrees 


480         640 


Fig.  9.     Starting  current  of  an  inductive  circuit. 

28.   Instead  of  considering,  in  Fig.  9,  the  current  wave  as 
consisting    of    the    superposition    of    the    permanent   term 

/  cos  (6  —  Q0)  and  the  transient  term  —  h     c  cos  00  the  current 
wave   can  directly   be   represented   by  the   permanent   term 


0 

4 

3 
2 
1 

0 
-1 

-2 
-3 

-4 
-ft 

^ 

\ 

X 

/ 

\ 

\ 
s.  > 

^ 

""S^ 

"N 

/ 

^~*. 

--^ 

\ 

^^. 

-  —  . 

/ 

/ 

\ 

1 

/ 

\ 

1 

\ 

/ 

\ 

/ 

V 

*/ 

Fig.  10.     Current  wave  represented  directly. 

I  cos  (0  -  00)  by  considering  the  zero  line  of  the  diagram  as 

-r-e 

deflected    exponentially  to  the   curve  h     ''    cos  00  in  Fig.  10. 
That  is,  the  instantaneous  values  of  current  are  the  vertical 


46  TRANSIENT  PHENOMENA 

distances  of  the  sine  wave  7  cos  (6  —  60)  from  the  exponential 

--fi 

curve    /s  cos  Q0,  starting  at  the  initial  value  of  perma- 

nent current. 
In  polar  coordinates,  in  this  case  7  cos  (0  —  00)  is  the  circle, 

-r-e 
h    x      cos  00  the  exponential  or  loxodromic  spiral. 

As  a  rule,  the  transient  term  in  alternating-current  circuits 
containing  resistance  and  inductance  is  of  importance  only  in 
circuits  containing  iron,  where  hysteresis  and  magnetic  saturation 
complicate  the  phenomenon,  or  in  circuits  where  unidirectional 
or  periodically  recurring  changes  take  place,  as  in  rectifiers, 
and  some  such  cases  are  considered  in  the  following  chapters. 


CHAPTER  V. 

RESISTANCE,    INDUCTANCE,  AND    CAPACITY    IN     SERIES. 
CONDENSER   CHARGE    AND    DISCHARGE. 

29.  If  a  continuous  e.m.f  .  e  is  impressed  upon  a  circuit  contain- 
ing resistance,  inductance,  and  capacity  in  series,  the  stationary 
condition  of  the  circuit  is  zero  current,  i  =  o,  and  the  poten- 
tial difference  at  the  condenser  equals  the  impressed  e.m.f., 
et  =•  e,  no  permanent  current  exists,  but  only  the  transient 
current  of  charge  or  discharge  of  the  condenser. 

The  capacity  C  of  a  condenser  is  defined  by  the  equation 

.          de 


that  is,  the  current  into  a  condenser  is  proportional  to  the  rate 
of  increase  of  its  e.m.f.  and  to  the  capacity. 
It  is  therefore 


and 

e-^-lidt  (1) 


is  the  potential  difference  at  the  terminals  of  a  condenser  of 
capacity  C  with  current  i  in  the  circuit  to  the  condenser. 

Let  then,  in  a  circuit  containing  resistance,  inductance,  and 
capacity  in  series,  e  =  impressed  e.m.f.,  whether  continuous, 
alternating,  pulsating,  etc.;  i  =  current  in  the  circuit  at  time  t; 
r  =  resistance;  L  =  inductance,  and  C  =  capacity;  then  the 
e.m.f.  consumed  by  resistance  r  is 

n; 

the  e.m.f.  consumed  by  inductance  L  is 

di 


47 


48  TRANSIENT  PHENOMENA 

and  the  e.m.f  .  consumed  by  capacity  C  is 


hence,  the  impressed  e.m.f.  is 


and  herefrom  the  potential  difference  at  the  condenser  terminals 
is 

Ci  =  -L   Cidt  =  e  -  ri  -  L  -*•  (3) 

(/  «/  at 

Equation  (2)  differentiated  and  rearranged  gives 
„  d?i         di      1  .      de 


as  the  general  differential  equation  of  a  circuit  containing  resist- 
ance, inductance,  and  capacity  in  series. 
30.  If  the  impressed  e.m.f.  is  constant, 

e  =  constant, 

de 

then  37  =  0, 

dt 

and  equation   (4)  assumes  the  form,   for    continuous-current 
circuits, 


This  equation  is  a  linear  relation  between  the  dependent  vari- 
able, i,  and  its  differential  quotients,  and  as  such  is  integrated 
by  an  exponential  function  of  the  general  form 

i  =  Ae-*.  (6) 

(This  exponential  function  also  includes  the  trigonometric 
functions  sine  and  cosine,  which  are  exponential  functions  with 
imaginary  exponent  a.) 


CONDENSER  CHARGE  AND  DISCHARGE  49 

Substituting  (6)  in  (5)  gives 


this  must  be  an  identity,  irrespective  of  the  value  of  t,  to  make 
(6)  the  integral  of  (5).     That  is, 

a?L-ar+-  =  Q.  (7) 

A  is  still  indefinite,  and  therefore  determined  by  the  terminal 
conditions  of  the  problem. 
From  (7)  follows 


£.  (8) 


hence  the  two  roots, 

r  —  s 


and 


r  + 


(9) 


where  s  =  y/r2  -  ^  .  (10) 

Since  there  are  two  roots,  at  and  a2,  either  of  the  two  expres- 
ions  (6),  e~ait  and  e~a2t,  and  therefore  also  any  combination  of 
these  two  expressions,  satisfies  the  differential  equation  (5). 

That  is,  the  general  integral  equation,  or  solution  of  differential 
equation  (5),  is 

i  =  Ai^  +  Ai^  .  (11) 


Substituting  (11)  and  (9)  in  equation  (3)  gives  the  potential 
difference  at  the  condenser  terminals  as 


e  — 


(12) 


50  TRANSIENT  PHENOMENA 

31.  Equations  (11)  and  (12)  contain  two  indeterminate  con- 
stants, A!  and  A2,  which  are  the  integration  constants  of  the 
differential  equation  of  second  order,  (5),  and  determined  by 
the  terminal  conditions,  the  current  and  the  potential  differ- 
ence at  the  condenser  at  the  moment  t  =  0. 

Inversely,  since  in  a  circuit  containing  inductance  and  capac- 
ity two  electric  quantities  must  be  given  at  the  moment  of 
start  of  the  phenomenon,  the  current  and  the  condenser  poten- 
tial —  representing  the  values  of  energy  stored  at  the  moment 
t  =  0  as  electromagnetic  and  as  electrostatic  energy,  respec- 
tively —  the  equations  must  lead  to  two  integration  constants, 
that  is,  to  a  differential  equation  of  second  order. 

Let  i  =  i0  =  current  and  et  =  e0  =  potential  difference  at 
condenser  terminals  at  the  moment  t  =  0;  substituting  in  (11) 
and  (12), 

t0  =  A,  +  A2 

and  e0  - 

hence, 


r  —  s  . 
~~2~l« 


and 


r  +  s 


(13) 


s 

and  therefore,  substituting  in  (11)  and  (12),  the  current  is 
r  +  s  .  r  —  s 

-^0    _rzf, 

~L   >       (14) 


_'+•,      «o-«-    -o-*« 


s  s 

the  condenser  potential  is 

r-t-s .  r—s. 


~    2L 


(15) 


CONDENSER  CHARGE  AND  DISCHARGE  51 

For  no  condenser  charge,  or  i0  =  0,  e0  =  0,  we  have 


e 

1      s 
and 


substituting  in  (11)  and  (12),  we  get  the  charging  current  as 

ii^/      _  r±»  *  ) 

2L    -•"    j.  (16) 

The  condenser  potential  as 


r+sH) 

2L    • 

J  ) 


For  a  condenser  discharge  or  i'0  =  0,  e  =  e0,  we  have 


and 

hence,  the  discharging  current  is 

(     _  r~s  r+s 

S    ( 

The  condenser  potential  is 


N 

' 
/ 


that  is,  in  condenser  discharge  and  in  condenser  charge  the 
currents  are  the  same,  but  opposite  in  direction,  and  the  con- 
denser potential  rises  in  one  case  in  the  same  way  as  it  falls  in 
the  other. 

32.   As  example  is  shown,  in  Fig.  11,  the  charge  of  a  con- 
denser .of  C  =  10    mf.   capacity   by  an   impressed  e.m.f.   of 


52 


TRANSIENT  PHENOMENA 


e  =  1000  volts  through  a  circuit  of  r  =  250  ohms  resistance 
and  L  =  100  mh.  inductance;  hence,  s  =  150  ohms,  and  the 
charging  current  is 

6.667  fe-500'  -  £-; 


i  =  D.DD/  [e  "       -  e~*  ""j  amperes. 
The  condenser  potential  is 

el  =  1000  {1  -  1.333  e~5<Mt  +  0.333  £-2000<j  volts. 


12        16       20       24       28       32        36       40 


Fig.  11.     Charging  a  condenser  through  a  circuit  having  resistance  and  induc 
tance.     Constant  potential.     Logarithmic  charge. 

33.  The  equations  (14)  to  (19)  contain  the  square  root, 


'4L 


hence,  they  apply  in  their  present  form  only  when 

4L 


If  r2  =  -—  -  ,  these  equations  become  indeterminate,  or  =  —  > 
0  0 

and  if  r2  <  —  ,  s  is  imaginary,  and  the  equations  assume  a 
C 

complex  imaginary   form.     In   either   case    they   have  to   be 
rearranged  to  assume  a  form  suitable  for  application. 
Three  cases  have  thus  to  be  distinguished  : 

(a)  r2  >  —  -,  in  which  the  equations  of  the  circuit  can  be 
o 

used  in  their  present  form.     Since  the  functions  are  exponen- 
tial or  logarithmic,  this  is  called  the  logarithmic  case. 


CONDENSER  CHARGE  AND  DISCHARGE  53 

(6)  r2  =  —    is  called  the  critical  case,  marking  the  transi- 
tion between  (a)  and  (c),  but  belonging  to  neither. 

(c)  r2  <  —  .    In  this  case  trigonometric  functions  appear;  it 

is  called  the  trigonometric  case,  or  oscillation. 
34.   In  the  logarithmic  case, 


4L<W, 

that  is,  with  high  resistance,  or  high  capacity,  or  low  induc- 
tance, equations  (14)  to  (19)  apply. 

*•-«  t    ^  r+s  t 

The  term  e  2L  is  always  greater  than  e  2L  ,  since  the 
former  has  a  lower  coefficient  in  the  exponent,  and  the  differ- 
ence of  these  terms,  in  the  equations  of  condenser  charge  and 
discharge,  is  always  positive.  That  is,  the  current  rises  from 
zero  at  t  =  0,  reaches  a  maximum  and  then  falls  again  to 

zero  at  £  =  oo,  but  it  never  reverses.    The  maximum  of  the 

/? 

current  is  less  than  i  =  -  • 

s 

The  exponential  term  in  equations  (17)  and  (19)  also  never 
reverses.  That  is,  the  condenser  potential  gradually  changes, 
without  ever  reversing  or  exceeding  the  impressed  e.m.f.  in  the 
charge  or  the  starting  potential  in  the  discharge. 

,  in  the  case  r2  >  —  —  ,  no  abnormal  voltage  is  pro- 


duced  in  the  circuit,  and  the  transient  term  is  of  short  duration, 
so  that  a  condenser  charge  or  discharge  under  these  conditions 
is  relatively  harmless. 

In  charging  or  discharging  a  condenser,  or  in  general  a  circuit 
containing  capacity,  the  insertion  of  a  resistance  in  series  in  the 

circuit    of  such  value  that  r2  >  —    therefore   eliminates    the 

C 

danger  from  abnormal  electrostatic  or  electromagnetic  stresses. 

In  general,  the  higher  the  resistance  of  a  circuit,  compared 
with  inductance  and  capacity,  the  more  the  transient  term  is 
suppressed. 


54  TRANSIENT  PHENOMENA 

35.    In  a  circuit  containing  resistance  and  capacity  but  no 
inductance,  L  =  0,  we  have,  substituting  in  (5), 

rf  +  jL;  =  0,  (20) 

or,  transposing, 


which  is  integrated  by  _t_ 

i  -  ce    rc,  (21) 

where  c  =  integration  constant. 

Equation  (21)  gives  for  t  =  0,  i  =  c;  that  is,  the  current  at 
the  moment  of  closing  the  circuit  must  have  a  finite  value,  or 
must  jump  instantly  from  zero  to  c.  This  is  not  possible,  but 
so  also  it  is  not  possible  to  produce  a  circuit  without  any  induc- 
tance whatever. 

Therefore  equation  (21)  does  not  apply  for  very  small  values 
of  time,  t,  but  for  very  small  t  the  inductance,  L,  of  the  circuit, 
however  small,  determines  the  current. 

The  potential  difference  at  the  condenser  terminals  from  (3)  is 

el  =  e  —  ri 
hence  t 

et  =  e  -  res    r°  (22) 

The  integration  constant  c  cannot  be  determined  from  equation 
(21)  at  t  =  0,  since  the  current  i  makes  a  jump  at  this  moment. 

But  from  (22)  it  follows  that  if  at  the  moment  t  =  0,  el  =  e0, 
e0  =  e  -  re, 

e-eQ 

hence,  c  =  -  -, 

r 

and  herefrom  the  equations  of  the  non-inductive  condenser 

circuit, 

_t_ 

f  _(6-e>"'<7  (23) 

r 

and  ' 

' 


As  seen,  these  equations  do  not  depend  upon  the  current  iQ  in 
the  circuit  at  the  moment  before  t  =  0. 


CONDENSER  CHARGE  AND  DISCHARGE  65 

36.   These  equations  do  not  apply  for  very  small  values  of  t, 
but  in  this  case  the  inductance,  L,  has  to  be  considered,  that  is, 
equations  (14)  to  (19)  used. 
For  L  =  0  the  second  term  in  (14)  becomes  indefinite,  as  it 

«, 

contains  e      ,  and  therefore  has  to  be  evaluated  as  follows: 
For  L  =  0,  we  have 

s  =  r. 


and 

r  — 


0 


and,  developed  by  the  binomial  theorem,  dropping  all  but  the 
first  term, 


r  —  s  =  • 

2L 


and 

r-s      1 

2L       rC' 
r  -f  s  _r_ 

2L     ~~L 

Substituting  these  values  in  equations  (14)  and  (15)  gives  the 
current  as 


r  r 

and  the  potential  difference  at  the  condenser  as 

_j_ 

€l  =  e  -  (e  -  e0)  e  *  ',  (26) 

that  is,  in  the  equation  of  the  current,  the  term 


56  TRANSIENT  PHENOMENA 

has  to  be  added  to  equation  (23)  .  This  term  makes  the  transition 
from  the  circuit  conditions  before  t  =  0  to  those  after  t  =  0, 
and  is  of  extremely  short  duration. 

For  instance,  choosing  the  same  constants  as  in  §  32,  namely  : 
e  =  1000  volts;  r  =  250  ohms;  C  =  10  mf.,  but  choosing  the 
inductance  as  low  as  possible,  L  =  5  mh.,  gives  the  equations 
of  condenser  charge,  i.e.,  for  iQ  =  0  and  e0  =  0, 


and 

e,  =  1000  {1  -  e-400'}. 

The  second  term  in  the  equation  of  the  current,  f-50-000',  has 
decreased  already  to  1  per  cent  after  t  =  17.3  X  10~6  seconds, 
while  the  first  term,  £-400';  has  during  this  time  decreased  only 
by  0.7  per  cent,  that  is,  it  has  not  yet  appreciably  decreased. 

37.   In  the  critical  case, 


•  c 

and  s  =  0,     . 

r 


A,  =  -  A 


s 
Hence,  substituting  in  equation  (14)  and  rearranging, 


O    7"  o    7" 


^=(e-eQ-r-i^-^(^     ^      -)•  (27) 

The  last  term  of  this  equation, 


s  s 

'-L  _,~"f    o 


D  s  'a' 


CONDENSER   CHARGE  AND  DISCHARGE  57 

that  is,    becomes   indeterminate   for  s  =  0,   and   therefore  is 
evaluated  by  differentiation, 


ds 
Substituting  (28)  in  (27)  gives  the  equation  of  current, 

^'-  (29) 

The  condenser  potential  is  found,  by  substituting  in  (15),  to  be 


(30) 


The  last  term  of  this  equation  (30)  is 


(31) 
For  s  =  0,  the  first  term  of  this  equation  (31),  by  substituting 

(28),  becomes  ^-~  ,  the  second  term  =  1,  and  substituting  in  (30), 
2  Li 

this  gives  the  condenser  potential  as 

^'.  (32) 


Herefrom  it  follows  that  for  the  condenser  charge,  iQ  =  0  and 
e0  =  0, 


=  Le 
and 

e.  =  e  ?!-(!  + 


2L 


58  TRANSIENT  PHENOMENA 

for  the  condenser  discharge,  i0  =  0  and  e  =  0, 


and 


38.  As  an  example  are  shown,  in  Fig.  12,  the  charging  current 
and  the  potential  difference  at  the  terminals  of  the  condenser, 


5—1000 


36        40 


Fig.  12.     Charging  a  condenser  through  a  circuit  having  resistance  and  induc- 
tance.    Constant  potential.     Critical  charge. 

in  a  circuit  having  the  constants',  e  =  1000  volts;  C  =  10  mf.; 
L  =  100  mh.,  and  such  resistance  as  to  give  the  critical  start, 
that  is, 


V  —  =  200  ohms. 
V   C 


In  this  case, 


and 


i  =  10,000  t£~looot 
e,  =  1000  {!-(!  +  10000  £"1000'}. 
39.   In  the  trigonometric  or  oscillating  case, 


The  term  under  the  square  root  (10)  is  negative,  that  is,  the 
square  root,  s,  is  imaginary,  and  al  and  a2  are  complex  imaginary 
quantities,  so  that  the  equations  (11)  and  (12)  appear  in  imagi- 
nary form.  They  obviously  can  be  reduced  to  real  terms, 


CONDENSER  CHARGE  AND  DISCHARGE 


59 


since  the  phenomenon  is  real.  Since  an  exponential  function 
with  imaginary  exponents  is  a  trigonometric  function,  and 
inversely,  the  solution  of  the  equation  thus  leads  to  trigono- 
metric functions,  that  is,  the  phenomenon  is  periodic  or  oscil- 
lating. 
Substituting  s  =  jq,  we  have 


(33) 


and 


Substituting  (34)  in  (11)  and  (12),  and  rearranging, 


^  = 


2L 


(34) 


(35) 


(36) 


Between   the   exponential    function   and   the   trigonometric 
functions  exist  the  relations 


and 


=  cos  v  +  j  sin  v 


y      _ 


(37) 


Substituting  (37)  in  (35),  and  rearranging,  gives 


-  (A,  +  A,)  cos^^  +  j  (A,  -  A2)  sin  -^«  j. 

Substituting  the  two  new  integration  constants, 

B,  =  AV  +  A2       ] 

and  (38) 

j^-7(At-A,),J 

gives 

(39) 


60  TRANSIENT  PHENOMENA 

In  the  same  manner,  substituting  (37)  in  (36),  rearranging, 
and  substituting  (38),  gives 


.t-.-™+**.   (40 

{  Zi  Zi  Ll  Zi  A  Ll      } 

Bl  and  B2  are  now  the  two  integration  constants,  determined 
by  the  terminal  conditions.  That  is,  for  t  =  0,  let  i  =  i0  =  cur- 
rent and  e^  =  e0  =  potential  difference  at  condenser  terminals, 
and  substituting  these  values  in  (39)  and  (40)  gives 


and 

rB,  -I-  qB, 


hence, 
and 


(41) 


Substituting  (41)  in  (39)  and  (40)  gives  the  general  equations 
of  condenser  oscillation: 
the  current  is 


and  the  potential  difference  at  condenser  terminals  is 

r  (e — ^O i 

0'  o  0 

'    (43) 

Herefrom  follow  the  equations  of  condenser  charge  and  dis- 
charge, as  special  case : 
For  condenser  charge,  iQ  =  0;     e0  =  0,  we  have 

A  *  (44) 


CONDENSER   CHARGE  AND  DISCHARGE 


and 


and  for  condenser  discharge,  i0  =  0,  e  =  0,  we  have 


and 


<V 


T    . 

-* 


61 

(45) 

(46) 

(47) 


40.  As  an  example  is  shown  the  oscillation  of  condenser 
charge  in  a  circuit  having  the  constants,  e  =  1000  volts;  L  = 
100  mh.,  and  C  =  10  mf. 


Fig.  13.     Charging  a  condenser  through  a  circuit  having  resistance  and  induc- 
tance.    Constant  potential.     Oscillating  charge. 

(a)  In  Fig.  13,  r  =  100  ohms,  hence,  q  =  173  and  the  current  is 

i  =  11.55  e-500t  sin  866  Z; 
the  condenser  potential  is 

e,  =  1000  { 1  -  e-  50° '  (cos  866  t  +  0.577  sin  866  0 } . 

(b)  In  Fig.  14,  r  =  40  ohms,  hence,  q  =  196  and  the  current 
is 

i  =  10.2  £-200'  sin  980*; 

the  condenser  potential  is 

e,  -  1000  { 1  -  e"  20° '  (cos  980  t  +  0.21  sin  980  0 } . 


62 


TRANSIENT  PHENOMENA 


41.  Since  the  equations  of  current  and  potential  difference 
(42)  to  (47)  contain  trigonometric  functions,  the  phenomena 
are  periodic  or  waves,  similar  to  alternating  currents.  They 

r 

differ  from  the  latter  by  containing  an  exponential  factor  e    2  L  , 
which  steadily  decreases  with  increase  of  t.    That  is,  the  sue- 


16UUI  — 

f 

^ 

f 

N, 

c  = 

«  1QOO  volts 

L  = 

=  1 

X)mh 

1 

X 

\ 

T  = 

40  oh, 

as 

C  = 

= 

OE 

af. 

\ 

^ 

~-v, 

.  

1 

\ 

\ 

k 

* 

— 

***** 

1 

\ 

^ 

\ 

^r 

, 

IE 

— 

De 

gre 

es 

XX 

3           or 

8 

0 

1& 

°\ 

2A 

fy 

3i 

20 

7400 

480' 

\5( 

^) 

& 

(0 

72 

^ 

-« 

1  —  ~, 

[_ 

\ 

1 

s 

^ 

S 

I/ 

J 

\ 

1 

\ 

/ 

\ 

/ 

Fig.  14.     Charging  a  condenser  through  a  circuit  having  resistance  and  induc- 
tance.    Constant  potential.     Oscillating  charge. 

cessive  half  waves  of  current  and  of  condenser  potential  pro- 
gressively decrease  in  amplitude.  Such  alternating  waves  of 
progressively  decreasing  amplitude  are  called  oscillating  waves. 

Since  equations  (42)  to  (47)  are  periodic,  the  time  t  can  be 
represented  by  an  angle  6,  so  that  one  complete  period  is  denoted 
by  2  n  or  one  complete  revolution, 

f)  *      4          O      -ft  (A  Q\ 

0  =  —t  =  2rft.  (48) 


hence,  the  frequency  of  oscillation  is 


or,  substituting 


gives  the  frequency  of  oscillation  as 


(49) 


(50) 


CONDENSER  CHARGE  AND  DISCHARGE 


68 


This  frequency  decreases  with  increasing  resistance  r,  and 

becomes  zero  for  (  —  —\  =  —  ,  that  is,  r2  =  —  ,  or  the  critical 
\2  LI       L/C  C 

case,  where  the  phenomenon  ceases  to  be  oscillating. 

If  the  resistance  is  small,  so  that  the  second  term  in  equa- 
tion (50)  can  be  neglected,  the  frequency  of  oscillation  is 


Substituting  0  for  t  by  equation  (48) 


in  equations  (42)  and  (43)  gives  the  general  equationSj 


_U 


2q 
r(e-  e0)- 


=  e  - 


and 


1  (e  -  e0)  cos 
d  =  2nft 

f-^-. 


sin  0 


JL      /JlV 
LC      V2L/  ' 


(52) 


,(53) 
(48) 


(50) 


42.   If  the  resistance  r  can  be  neglected,  that  is,  if  r2  is  small 
compared  with  —  ,  the  following  equations  are  approximately 


exact: 

and 


r  _  __  _  _ 
J  '    2  n  VW 


(54) 


(55) 


64  TRANSIENT  PHENOMENA 

Introducing    now    x  =  2  TT/L  =  inductive    reactance    and 

x'  = ^-  =  capacity  reactance,   and    substituting    (55),   we 

2  TT/ C 

have 


and 

0 

hence,  xf  =  x, 

that  is,  the  frequency  of  oscillation  of  a  circuit  containing 
inductance   and    capacity,    but   negligible    resistance,    is    that 

frequency  /  which  makes  the  condensive  reactance  xf  =  — — — 

2  7T/C 

equal  the  inductive  reactance  x  =  2  nfL : 


(56) 

Then  (54), 

q  =  2x,  (57) 

and  the  general  equations  (52)  and  (53)  are 


i  =  s    2ar  ]  i0cos^  +  -  -  sin/9  [  ;  (58) 

4  ic 


-e0)  cos  0  +  r  g~c*~2a"Bin  0  1  •  (59) 
^  x 


x-  (56) 

and  by  (48)  and  (55)  : 


CONDENSER  CHARGE  AND  DISCHARGE 


65 


43.   Due  to  the  factor  e      '  ,  successive  half  waves  of  oscilla- 
tion decrease  the  more  in  amplitude,  the  greater  the  resistance  r. 
The  ratio  of  the  amplitude  of  successive  half  waves,  or  the 


decrement  of  the  oscillation,  is  A  =  e   2L  \  where  tl  =  duration 

of  one  half  wave  or  one  half  cycle,  =  -—  . 

2/ 


A 

a.o 


0.1        0.2         0.3        0.4         0.5        0.6         0.7        0.8        0.9        1.0 

Fig.  15.     Decrement  of  Oscillation. 


Hence,  from  (50), 


and 


Denoting  the  critical  resistance  as 


2    _ 


4L 


r>  ::  c' 


we  have 


or, 


A  «e 


-s 


-^-i 


(60) 
(61) 


(62) 


66  TRANSIENT  PHENOMENA 

that  is,  the  decrement  of  the  oscillating  wave,  or  the  decay  of 
the  oscillation,  is  a  function  only  of  the  ratio  of  the  resistance 
of  the  circuit  to  its  critical  resistance,  that  is,  the  minimum 
resistance  which  makes  the  phenomenon  non-oscillatory. 

In  Fig.  15  are  shown  the  numerical  values  of  the  decrement  A, 

for  different  ratios  of  actual  to  critical   resistance  —  • 

ri 

As  seen,  for  r  >  0.21  rv  or  a  resistance  of  the  circuit  of  more 
than  21  per  cent  of  its  critical  resistance,  the  decrement  A  is 
below  50  per  cent,  or  the  second  half  wave  less  than  half  the  first 
one,  etc. ;  that  is,  very  little  oscillation  is  left. 

Where  resistance  is  inserted  into  a  circuit  to  eliminate  the 
danger  from  oscillations,  one-fifth  of  the  critical  resistance,  or 

r  =  0.4  y—,  seems  sufficient  to  practically  dampen  out  the 
G 

oscillation. 


CHAPTER  VI. 

OSCILLATING  CURRENTS, 

44.  The  charge  and  discharge  of  a  condenser  through  an 
inductive  circuit  produces  periodic  currents  of  a  frequency 
depending  upon  the  circuit  constants. 

The  range  of  frequencies  which  can  be  produced  by  electro- 
dynamic  machinery  is  rather  limited:  synchronous  machines 
or  ordinary  alternators  can  give  economically  and  in  units  of 
larger  size  frequencies  from  10  to  125  cycles.  Frequencies 
below  10  cycles  are  available  by  commutating  machines  with 
low  frequency  excitation.  Above  125  cycles  the  difficulties 
rapidly  increase,  due  to  the  great  number  of  poles,  high  periph- 
eral speed,  high  power  required  for  field  excitation,  poor  regu- 
lation due  to  the  massing  of  the  conductors,  which  is  required 
because  of  the  small  pitch  per  pole  of  the  machine,  etc.,  so  that 
1000  cycles  probably  is  the  limit  of  generation  of  constant 
potential  alternating  currents  of  appreciable  power  and  at  fair 
efficiency.  For  smaller  powers,  a  few  kilowatts,  by  using 
shunted  capacity  to  assist  the  excitation,  and  not  attempting 
to  produce  constant  potential,  single-phase  alternators  have 
been  built  and  are  in  commercial  service  giving  10,000  and  even 
100,000  cycles,  and  200,000-cycle  alternators  are  being  designed 
for  wireless  telegraphy  and  telephony. 

Still,  even  going  to  the  limits  of  peripheral  speed,  and  sacri- 
ficing everything  for  high  frequency,  a  limit  is  reached  in  the 
frequency  available  by  electrodynamic  generation. 

It  becomes  of  importance,  therefore,  to  investigate  whether 
by  the  use  of  the  condenser  discharge  the  range  of  frequencies 
can  be  extended. 

Since  the  oscillating  current  approaches  the  effect  of  an 
alternating  current  only  if  the  damping  is  small,  that  is,  the 
resistance  low,  the  condenser  discharge  can  be  used  as  high 
frequency  generator  only  by  making  the  circuit  of  as  low  resist- 
ance as  possible. 

67 


68  TRANSIENT  PHENOMENA 

This,  however,  means  limited  power.  When  generating  oscillat- 
ing currents  by  condenser  discharge,  the  load  put  on  the  circuit, 
that  is,  the  power  consumed  in  the  oscillating-current  circuit, 
represents  an  effective  resistance,  which  increases  the  rapidity 
of  the  decay  of  the  oscillation,  and  thus  limits  the  power,  and, 
when  approaching  the  critical  value,  also  lowers  the  frequency. 
This  is  obvious,  since  the  oscillating  current  is  the  dissipation 
of  the  energy  stored  electrostatically  in  the  condenser,  and  the 
higher  the  resistance  of  the  circuit,  the  more  rapidly  is  this 
energy  dissipated,  that  is,  the  faster  the  oscillation  dies  out. 

With  a  resistance  of  the  circuit  sufficiently  low  to  give  a  fairly 
well  sustained  oscillation,  the  frequency  is,  with  sufficient 
approximation, 


45.  The  constants,  capacity,  C,  inductance,  L,  and  resistance,  r, 
have  no  relation  to  the  size  or  bulk  of  the  apparatus.  For 
instance,  a  condenser  of  1  mf.,  built  to  stand  continuously  a 
potential  of  10,000  volts,  is  far  larger  than  a  200-volt  condenser 
of  100  mf.  capacity.  The  energy  which  the  former  is  able  to 

Ce2 
store  is  -77-=  50  joules,  while  the  latter  stores  only  2  joules, 

2 

and  therefore  the  former  is  25  times  as  large. 
A  reactive  coil  of  0.1  henry  inductance,  designed  to  carry 

continuously  100  amperes,  stores—  =  500  joules;  a  reactive 

coil  of  1000  times  the  inductance,  100  henrys,  but  of  a  current- 
carrying  capacity  of  1  ampere,  stores  5  joules  only,  therefore  is 
only  about  one-hundredth  the  size  of  the  former. 

A  resistor  of  1  ohm,  carrying  continuously  1000  amperes,  is  a 
ponderous  mass,  dissipating  1000  kw.;  a  resistor  having  a 
resistance  a  million  times  as  large,  of  one  megohm,  may  be  a  lead 
pencil  scratch  on  a  piece  of  porcelain. 

Therefore  the  size  or  bulk  of  condensers  and  reactors  depends 
not  only  on  C  and  L  but  also  on  the  voltage  and  current  which 
can  be  applied  continuously,  that  is,  it  is  approximately  pro- 

Ce2        W 
portional  to  the  energy  stored,     -  and  —  ,  or  since  in  electrical 


OSCILLATING  CURRENTS  69 

engineering  energy  is  a  quantity  less  frequently  used  than 
power,  condensers  and  reactors  are  usually  characterized  by 
the  power  or  rather  apparent  power  which  can  be  impressed 
upon  them  continuously  by  referring  to  a  standard  frequency, 
for  which  60  cycles  is  generally  used. 

That  means  that  reactors,  condensers,  and  resistors  are  rated 
in  kilowatts  or  kilo  volt-amperes,  just  as  other  electrical  appa- 
ratus, and  this  rating  characterizes  their  size  within  the  limits 
of  design,  while  a  statement  like  "a  condenser  of  10  mf. "  or 
"a  reactor  of  100  mh."  no  more  characterizes  the  size  than  a 
Statement  like  "an  alternator  of  100  amperes  capacity"  or  "a 
transformer  of  1000  volts. " 

A  bulk  of  1  cu.  ft.  in  condenser  can  give  about  5  to  10 
kv-amp.  at  60  cycles.  Hence,  100  kv-amp.  constitutes  a  very 
large  size  of  condenser. 

In  the  oscillating  condenser  discharge,  the  frequency  of  oscil- 
lation is  such  that  the  inductive  reactance  equals  the  condensive 
reactance.  The  same  current  is  in  both  at  the  same  terminal 
voltage.  That  means  that  the  volt-amperes  consumed  by  the 
inductance  equal  the  volt-amperes  consumed  by  the  capacity. 

The  kilovolt-amperes  of  a  condenser  as  well  as  of  a  reactor 
are  proportional  to  the  frequency.  With  increasing  frequency, 
at  constant  voltage  impressed  upon  the  condenser,  the  current 
varies  proportionally  with  the  frequency;  at  constant  alter- 
nating current  through  the  reactor,  the  voltage  varies  propor- 
tionally with  the  frequency. 

If  then  at  the  frequency  of  oscillation,  reactor  and  con- 
denser have  the  same  kv-amp.,  they  also  have  the  same  at 
60  cycles. 

A  100-kv-amp.  condenser  requires  a  100-kv-amp.  reactive 
coil  for  generating  oscillating  currents.  A  100-kv-amp.  react- 
ive coil  has  approximately  the  same  size  as  a  50-kw.  trans- 
former and  can  indeed  be  made  from  such  a  transformer,  of 
ratio  1  :  1,  by  connecting  the  two  coils  in  series  and  inserting 
into  the  magnetic  circuit  an  air  gap  of  such  length  as  to  give 
the  rated  magnetic  density  at  the  rated  current. 

A  very  large  oscillating-current  generator,  therefore,  would 
consist  of  100-kv-amp.  condenser  and  100-kv-amp.  reactor. 

46.  Assuming  the  condenser  to  be  designed  for  10,000  volts 
alternating  impressed  e.m.f.  at  60  cycles,  the  100  kv-amp.  con- 


70  TRANSIENT  PHENOMENA 

denser   consumes    10   amperes:   its    condensive    reactance    is 

F  1 

xc  —  —  =  1000  ohms,  and  the  capacity  C=  — —  =  2.65  mf . 

I  2  7tJ0Xc 

Designing  the  reactor  for  different  currents,  and  therewith 
different  voltages,  gives  different  values  of  inductance  L,  and 
therefore  of  frequency  of  oscillation  /. 

From  the  equations  of  the  instantaneous  values  of  the  con- 
denser discharge,  (46)  and  (47),  follow  their  effective  values,  or 
Vmean  square, 


, - 

\/2 


and 


(63) 


q 

and  thus  the  power, 

Pi  =e1i  =  e-±\fe£~'Lt,  (64) 

since  for  small  values  of  r 


Herefrom  would  follow  that  the  energy  of  each  discharge  is 

W  =  Jo  p,dt  =  ^-  VOL.  (65) 

Therefore,  for  10,000  volts  effective  at  60  cycles  at  the  con- 
denser terminals,  the  e.m.f.  is 

e0  =  ' 10,000  \/2, 
and  the  condenser  voltage  is 

e,  =  10,000  r^'. 

Designing   now  the   100-kv-amp.  reactive  coil  for  different 
voltages  and  currents  gives  for  an  oscillation  of  10,000  volts : 


OSCILLATING  CURRENTS 


71 


Reactive  Coil. 

React- 
ance. 

Inductance. 

Frequency 
of 
Oscillation. 

Oscillating 
Current. 

Oscillating 
Power. 

Amp. 

Volts, 

^=*o- 

xn      L 

/           1 

Amp., 

Kv-amp., 

V 

e0. 

*o 

271/0 

In^LC 

i. 

Pi- 

1 

100,000 

105 

265 

6 

1 

10 

10 

10,000 

103 

2.65 

60 

10 

100 

100 

1,000 

10 

2.65xlO-2 

600 

100 

1,000 

1,000 

100 

10-  » 

2.65xlO-4 

6,000 

1,000 

10,000 

10,000 

10 

10-  3 

2.65xlO-8 

60,000 

10,000 

100,000 

100,000 

1 

10-  5 

2.65X10-8 

600,000 

100,000 

1,000,000 

-fz* 

r 

As  seen,  with  the  same  kilovolt-ampere  capacity  of  con- 
denser and  of  reactive  coil,  practically  any  frequency  of  oscil- 
lation can  be  produced,  from  low  commercial  frequencies  up  to 
hundred  thousands  of  cycles. 

At  frequencies  between  500  and  2000  cycles,  the  use  of  iron  in 
the  reactive  coil  has  to  be  restricted  to  an  inner  core,  and  at 
frequencies  above  this  iron  cannot  be  used,  since  hysteresis 
and  eddy  currents  would  cause  excessive  damping  of  the  oscil- 
lation. The  reactive  coil  then  becomes  larger  in  size. 

47.  Assuming  96  per  cent  efficiency  of  the  reactive  coil  and 
99  per  cent  of  the  condenser, 


gives 


since 


r  =  0.05  x, 

r  -  0.05  V 

x  =  2  xfL, 

1 


and  the  energy  of  the  discharge,  by  (65),  is 


W  =  —  -  \^LC  =  10  6*  C  volt-ampere-seconds; 

—  T 


thus  the  power  factor  is 


cos  00  =  0.05. 


72  .  TRANSIENT  PHENOMENA 

Since  the  energy  stored  in  the  capacity  is 

WQ  =  ^  joules, 
the  critical  resistance  is 


hence, 


r.      -  „  0 


7  =  0.025, 

*'4 


and  the  decrement  of  the  oscillation  is 

A  =  0.92, 

that  is,  the  decay  of  the  wave  is  very  slow  at  no  load. 

Assuming,  however,  as  load  an  external  effective  resistance 
equal  to  three  times  the  internal  resistance,  that  is,  an  elec- 
trical efficiency  of  75  per  cent,  gives  the  total  resistance  as 

r  +  r'  =  0.2  x\ 
hence, 


and  the  decrement  is 

A  =  0.73; 

hence  a  fairly  rapid  decay  of  the  wave. 

At  high  frequencies,  electrostatic,  inductive,  and  radiation 
losses  greatly  increase  the  resistance,  thus  giving  lower  effi- 
ciency and  more  rapid  decay  of  the  wave. 

48.  The  frequency  of  oscillation  does  not  directly  depend 
upon  the  size  of  apparatus,  that  is,  the  kilovolt-ampere  capacity 
of  condenser  and  reactor.  Assuming,  for  instance,  the  size,  in 

kilo  volt-amperes,  reduced  to  —  ,  then,  designed  for  the  same 

voltage,  condenser  and  reactor,  each  takes  —  the  current,  that 

n 

is,  the  condensive  reactance  is  n  times  as  great,  and  therefore 
the  capacity  of  the  condenser,  C,  reduced  to  —  ,  the  inductance,  L, 


OSCILLATING  CURRENTS  73 

is  increased  n-fold,  so  that  the  product  CL,  and  thereby  the 
frequency,  remains  the  same;  the  power  output,  however,  of  the 

oscillating  currents  is  reduced  to—. 

n 

The  limit  of  frequency  is  given  by  the  mechanical  dimensions. 

With  a  bulk  of  condenser  of  10  to  20  cu.  ft.,  the  minimum 
length  of  the  discharge  circuit  cannot  well  be  less  than  10  ft. ; 
10  ft.  of  conductor  of  large  size  have  an  inductance  of  at  least 
0.002  mh.  =  2  X  10  ~6,  and  the  frequency  of  oscillation  would 
therefore  be  limited  to  about  60,000  cycles  per  second,  even 
without  any  reactive  coil,  in  a  straight  discharge  path. 

The  highest  frequency  which  can  be  reached  may  be  estimated 
about  as  follows : 

The  minimum  length  of  discharge  circuit  is  the  gap  between 
the  condenser  plates. 

The  minimum  condenser  capacity  is  given  by  two  spheres, 
since  small  plates  give  a  larger  capacity,  due  to  the  edges. 

The  minimum  diameter  of  the  spheres  is  1.5  times  their 
distance,  since  a  smaller  sphere  diameter  does  not  give  a  clean 
spark  discharge,  but  a  brush  discharge  precedes  the  spark. 

With  e0  =  10,000  V2,  the  spark  gap  length  between  spheres 
is  e=  0.3  in.,  and  the  diameter  of  the  spheres  therefore  0.45  in. 
The  oscillating  circuit  then  consists  of  two  spheres  of  0.45  in., 
separated  by  a  gap  of  0.3  in. 

This  gives  an  approximate  length  of  oscillating  circuit  of 

0  3  X  10~3 
0.5  in.,  or  an  inductance  L  =  =0.125  X  10"7  henry. 

The  capacity  of  the  spheres  against  each  other  may  be 
estimated  as  C  =  50  X  10" 8  mf.;  this  gives  the  frequency  of 
oscillation  as  , 

/  = j=  =  2  X  109, 

or,  2  billion  cycles. 
At  e0  =  10,000  V2  volts, 

e,  =  10,000  e     2L    volts, 


-  —  t 
i  =  2.83  £   2L   amp., 


and  pl  =  28.3  s    L    kv-amp. 


74  TRANSIENT  PHENOMENA 

Reducing  the  size  and  spacing  of  the  spheres  proportionally, 
and  proportionally  lowering  the  voltage,  gives  still  higher 
frequencies. 

As  seen,  however,  the  power  of  the  oscillation  decreases  with 
increasing  frequency,  due  to  the  decrease  of  size  and  therewith 
of  storage  ability,  of  capacity,  and  of  inductance. 

With  a  frequency  of  billions  of  cycles  per  second,  the  effective 
resistance  must  be  very  large,  and  therefore  the  damping  rapid. 

Such  an  oscillating  system  of  two  spheres  separated  by  a  gap 
would  have  to  be  charged  by  induction,  or  the  spheres  charged 
separately  and  then  brought  near  each  other,  or  the  spheres 
may  be  made  a  part  of  a  series  of  spheres  separated  by  gaps  and 
connected  across  a  high  potential  circuit,  as  in  some  forms  of 
lightning  arresters. 

Herefrom  it  appears  that  the  highest  frequency  of  oscillation 
of  appreciable  power  which  can  be  produced  by  a  condenser 
discharge  reaches  billions  of  cycles  per  second,  thus  is  enormously 
higher  than  the  highest  frequencies  which  can  be  produced  by 
electrodynamic  machinery. 

At  five  billion  cycles  per  second,  the  wave  length  is  about 
6  cm.,  that  is,  the  frequency  only  a  few  octaves  lower  than 
the  lowest  frequencies  observed  as,  heat  radiation  or  ultra  red 
light. 

The  average  wave  length  of  visible  light,  55  X  10~6  cm., 
corresponding  to  a  frequency  of  5.5  X  1014  cycles,  would  require 
spheres  10~5  cm.  in  diameter,  that  is,  approaching  molecular 
dimensions. 

OSCILLATING-CURRENT   GENERATOR. 

49.  A  system  of  constant  impressed  e.m.f.,  e,  charging  a  con- 
denser C  through  a  circuit  of  inductance  L  and  resistance  r,  with 
a  discharge  circuit  of  the  condenser,  C,  comprising  an  air  gap 
in  series  with  a  reactor  of  inductance  L0  and  a  resistor  of  resist- 
ance r0,  is  a  generator  of  oscillating  current  if  the  air  gap  is  set 
for  such  a  voltage  e0  that  it  discharges  before  the  voltage  of  the 
condenser  C  has  reached  the  maximum,  and  if  the  resistance  r0 
is  such  as  to  make  the  condenser  discharge  oscillatory,  that  is, 


OSCILLATING  CURRENTS 


75 


In  such  a  system,  as  shown  diagrammatically  in  Fig.  16,  as 
soon,  during  the  charge  of  the  condenser,  as  the  terminal  voltage 
at  C  and  thereby  at  the  spark  gap  has  reached  the  value  e0,  the 
condenser  C  discharges  over  this  spark  gap,  its  potential  dif- 
ference falls  to  zero,  then  it  charges  again  up  to  potential  differ- 


ence eQ,  discharges,  etc.    Thus  a  series  of  oscillating  discharges 


Fig.  16.     Oscillating-current  generator. 

occur  in  the  circuit,  L0,  r0,  at  intervals  equal  to  the  time  required 
to  charge  condenser  C  over  reactor  L  and  resistor  r,  up  to  the 
potential  difference  e0,  with  an  impressed  e.m.f.  e. 

The  resistance,  r,  obviously  should  be  as  low  as  possible,  to 
get  good  efficiency  of  transformation;  the  inductance,  L,  must 
be  so  large  that  the  time  required  to  charge  condenser  C  to 
potential  e0  is  sufficient  for  the  discharge  over  L0,  r0  to  die  out 
and  also  the  spark  gap  e0  to  open,  that  is,  the  conducting  products 
of  the  spark  in  the  gap  e0  to  dissipate.  This  latter  takes  a  con- 
siderable time,  and  an  air  blast  directed  against  the  spark  gap  e0, 
by  carrying  away  the  products  of  the  discharge,  permits  a  more 
rapid  recurrence  of  the  discharge.  The  velocity  of  the  air  blast 
(and  therefore  the  pressure  of  the  air)  must  be  such  as  to  carry 
the  ionized  air  or  the  metal  vapors  which  the  discharge  forms 
in  the  gap  e0  out  of  the  discharge  path  faster  than  the  con- 
denser recharges. 

Assuming,  for  instance,  the  spark  gap,  e0,  set  for  20,000  volts, 
or  about  0.75  in.,  the  motion  of  the  air  blast  during  successive 
discharges  then  should  be  large  compared  with  0.75  in.,  hence 
at  least  3  to  6  in.  With  1000  discharges  per  second,  this  would 
require  an  air  velocity  of  v  =  250  to  500  feet  per  second,  with 
5000  discharges  per  second  an  air  velocity  of  v  =  1250  to  2500 
feet  per  second,  corresponding  to  an  air  pressure  of  approximately 
p  =  14.7  { (1  +  2  w2 10  - 7)3'5  -  1 }  lb.  per  sq.  in.,  or  0.66  to  2.75 
Ib.  in  the  first,  23  to  230  lb.  in  the  second  case. 


76  TRANSIENT  PHENOMENA 

While  the  condenser  charge  may  be  oscillatory  or  logarithmic, 
efficiency  requires  a  low  value  of  r,  that  is,  an  oscillatory  charge. 

With  a  frequency  of  discharge  in  L0,  r0  very  high  compared 
with  the  frequency  of  charge,  the  duration  of  the  discharge  is 
short  compared  with  the  duration  of  the  charge,  that  is,  the 
oscillating  currents  consist  of  a  series  of  oscillations  separated 
by  relatively  long  periods  of  rest.  Thus  the  current  in  L  does 
not  appreciably  change  during  the  time  of  the  discharge,  and  at 
the  end  of  the  condenser  charge  the  current  in  the  reactor,  L, 
is  the  same  as  the  current  in  L,  with  which  the  next  condenser 
charge  starts.  The  charging  current  of  the  condenser,  C,  in  L 
thus  changes  from  iQ  at  the  beginning  of  the  charge,  or  con- 
denser e.m.f.,  e0  =  0,  to  the  same  value  i0  at  the  end  of  the 
charge,  or  condenser  e.m.f.,  e±  =  e0. 

50.  Counting,  therefore,  the  time,  t,  from  the  moment  when 
the  condenser  charge  begins,  we  have  the  terminal  conditions  : 

t  =  0,  i  =  iQ}  e1  =  0  at  the  beginning  of  the  condenser  charge. 
t  =  t0,    i  =  i0,     e^  =  e0    at  the  end  of  the  condenser  charge. 

In  the  condenser  discharge,  through  circuit  Lo;  r0,  counting 
the  time  t'  from  the  moment  when  the  condenser  discharge 
begins,  that  is,  t'  =  t  —  t0,  we  have 

t'  =  0,    i  =  0,    et  =  e0  the  terminal  condition. 

e0,  thus,  is  that  value  of  the  voltage  e±  at  which  discharge 
takes  place  across  the  spark  gap,  and  t0  is  the  time  elapsing 
between  et  =  0  and  e1  =  e0,  or  the  time  required  to  build  Up 
the  voltage  e±  sufficiently  to  break  down  the  spark  gap. 

Under  the  assumption  that  the  period  of  oscillation  of  the 
condenser  charge  through  L,  r,  is  large  compared  with  the 
period  of  oscillation  of  the  condenser  discharge  through  L0,  r0, 
the  equations  are: 

(A)  Condenser  discharge  : 

*~^rA'tfoJ^*>,  (66) 

(67) 


where 

(68) 


OSCILLATING  CURRENTS 
(B)  Condenser  charge : 

i  =  £ 


V  =   c     2  L 


2  e  -  rin 


11 


'     (69) 


r2  +  cf 
re —  i 

q    .   ,  2        °    .       q 

_<+.  .^        t 


where 


,     (70) 


(71) 


Substituting  in  (69)  and  (70)  the  above  discussed  terminal 
conditions, 

it  *  * 

gives 

r    .    /  n  (y       •  » 

V    -.    -~2Z<0)y   «n«     "    >    4._  °dn    ?    /    (  f79^ 

in    —     *  \   ''0  ^"O  ~ — ~     ^0      '     ~  Dill          _     t()    /  If  &) 

(  2  L  q  2  L      ) 

and 


+ 


re  — 


e- 


e  cos  - 


Denoting,  for  convenience, 
r 


2L  ° 


and 


-  =  a, 


and  resolving  (72)  for  i'0,  gives 
2e 


q    1  —  e~8  cos  <^>  +  as  8  sm(f> 
and  substituting  (75)  in  (73)  and  rearranging, 


e0  =  e 


(73) 


(74) 


(75) 


(76) 


78 


TRANSIENT  PHENOMENA 


The  two  equations  (75),  (76)  permit  the  calculation  of  two  of 
the  three  quantities  i0,  ew  t0:  the  time,  t0,  of  condenser  charge 
appears  in  the  exponential  function,  in  s,  and  in  the  trigonometric 
function,  in  <j>. 

Since  in  an  oscillating-current  generator  of  fair  efficiency, 
that  is,  when  r  is  as  small  as  possible,  s  is  a  small  quantity, 
£-s  can  be  resolved  into  the  series 


£~S  =  1  ~  S  +  -  -  + 


(77) 


Substituting  (77)  in  (75),  and  dropping  all  terms  higher  than 
,  gives 


(S2\ 
1  -  s  +  -  j  sin  <f> 


q  s2 

1  —  cos  (f)  +  s  cos  9  —  —  cos  <p  +  a  sm  </>  —  as  sin  cf> 

a 

Multiplying  numerator   and    denominator    by  f  1  +  -J,   and 


rearranging,  gives 


2e 


sn. 


g2  +  s 
2-  s 

2e 


—  cos  (/>  +  a  sin 


sn 


+  2  sin2  ~  +  a  sin 

~j 


(78) 


Substituting  (77)  in  (76),  dropping  terms  higher  than  s2  and 
as,  multiplying  numerator  and  denominator  by  f  1  +-«Ji  and 


rearranging,  gives 


2s  .  ,  6 

+  2  sm2  —  +  a  sin 


(79) 


2  - 


OSCILLATING  CURRENTS  79 

Substituting  t0  in  (78)  and  (79)  gives 


i--  ' 

' 


4L-rt0  4L  °      q       2  L  ° 

and 


as  approximate  equations  giving  iQ  and  e0  as  functions  of  t0,  or  the 
time  of  condenser  charge. 

51.  The  time,  t0,  during  which  the  condenser  charges,  increases 
with  increasing  e0,  that  is,  increasing  length  of  the  spark  gap  in 
the  discharge  circuit,  at  first  almost  proportionally,  then,  as 
eo  approaches  2  e,  more  slowly. 

As  long  as  e0  is  appreciably  below  2  e,  that  is,  about  e0  <  1.75  e, 
t0  is  relatively  short,  and  the  charging  current  i,  which  increases 
from  i0  to  a  maximum,  and  then  decreases  again  to  i0,  does  not 
vary  much,  but  is  approximately  constant,  with  an  average 
value  very  little  above  iot  so  that  the  power  supplied  by  the 
impressed  e.m.f.,  e,  to  the  charging  circuit  can  approximately 
be  assumed  as 

P.  =  «V  (82) 

The  condenser  discharge  is  intermittent,  consisting  of  a  series 
of  oscillations,  with  a  period  of  rest  between  the  oscillations, 
which  is  long  compared  with  the  duration  of  the  oscillation, 
and  during  which  the  condenser  charges  again. 

The  discharge  current  of  the  condenser  is,  (66), 

' 


and  since  such  an  oscillation  recurs  at  intervals  of  t0  seconds, 

the  effective  value,  or  square  root  of  mean  square  of  the  dis- 
charge current,  is 

-dt.  (83) 


80  TRANSIENT  PHENOMENA 

Long  before  t  =  t0,  i  is  practically  zero,  and  as  upper  limit  of 
the  integral  can  therefore  be  chosen  <x>  instead  of  t0. 

Substituting  (66)  in  (83),  and  taking  the  constant  terms  out 
of  the  square  root,  gives  the  effective  value  of  discharge  cur- 
rent as 


however. 


and  by  fractional  integration, 
J  s'^'coa^tdt 


00 


1  +  p- 


hence,  substituting  in  (84), 


Since 


we  have,  substituting  in  (85), 

f~r~ 
V-e.y^L.,  (86) 

and,  denoting  by 


OSCILLATING  CURRENTS  81 

the  frequency  of  condenser  charge,  or  the  number  of  complete 
trains  of  discharge  oscillations  per  second, 

(87) 

that  is,  the  effective  value  of  the  discharge  current  is  propor- 
tional to  the  condenser  potential,  e0,  proportional  to  the  square 
root  of  the  capacity,  C,  and  the  frequency  of  charge,  fv  and 
inversely  proportional  to  the  square  root  of  the  resistance,  r0, 
of  the  discharge  circuit;  but  it  does  not  depend  upon  the  induc- 
tance L0  of  the  discharge  circuit,  and  therefore  does  not  depend 
on  the  frequency  of  the  discharge  oscillation. 
The  power  of  the  discharge  is 

Pi-tTr.-/,^-  (88) 

e  2C 
Since  -2—  is  the  energy  stored  in  the  condenser  of  capacity  C 

at  potential  e0,  and  /t  the  frequency  or  number  of  discharges 
of  this  energy  per  second,  equation  (88)  is  obvious. 

Inversely  therefore,  from  equation  (88),  that  is,  the  total 
energy  stored  in  the  condenser  and  discharging  per  second, 
the  effective  value  of  discharge  current  can  be  directly  calcu- 
lated as 


The  ratio  of  effective  discharge  current,  iv  to  mean  charging 
current,  i0,  is 


and  substituting  (80)  and  (81)  in  (89), 

3  *  2 

l4L>     8L-  (90) 


82  TRANSIENT  PHENOMENA 

The  magnitude  of  this  quantity  can  be  approximated  by 

neglecting  r  compared  with  -^-,  that  is,  substituting  q  =  y  — 

C  *     G 

and  replacing  the  sine-function  by  the  arcs.     This  gives 


that  is,  the  ratio  of  currents  is  inversely  proportional  to  the 
square  root  of  the  resistance  of  the  discharge  circuit,  of  the 
capacity,  and  of  the  frequency  of  charge. 

52.  Example:  Assume  an  oscillating-current  generator,  feed- 
ing a  Tesla  transformer  for  operating  X-ray  tubes,  or  directly 
supplying  an  iron  arc  (that  is,  a  condenser  discharge  between 
iron  electrodes)  for  the  production  of  ultraviolet  light. 

The  constants  of  the  charging  circuit  are:  the  impressed 
e.m.f.,  e  =  15,000  volts;  the  resistance,  r  =  10,000  ohms;  the 
inductance,  L  =  250  henrys,  and  the  capacity,  C  =  2  X  10~  8 
farads  =  0.02  mf. 

The  constants  of  the  discharge  circuit  are:  (a)  operating 
Tesla  transformer,  the  estimated  resistance,  r0  =  20  ohms 
(effective)  and  the  estimated  inductance,  L0  =  60  X  10"  fl 
henry  =  0.06  mh.;  (b)  operating  ultraviolet  arc,  the  esti- 
mated resistance,  r0  =  5  ohms  (effective)  and  the  estimated 
inductance,  L0  =  4  X  10"  6  henry  =  0.004  mh. 

Therefore  in  the  charging  circuit, 

q  =  223,400  ohms,    '  0.0448, 


=446.8, 


2L  2L 


-  =     0.025; 


t.  =  0.1344 sin  446.8  <0 


0      +  2  sin2  223.4  <0+  0.0448  sin  446.8  ta 
U.J.  —  CQ 

2  \  (92) 

30,000-  — ^—       — *— 

2^      +  2 sin2 223.4  Z0+  0.0448 sin  446.8  t. 


OSCILLATING  CURRENTS  83 

Fig.  17  shows  iQ  and  e0  as  ordinates,  with  the  time  of  charge 
tQ  as  abscissas. 


<0-=0.2     0.4     0.6     0.8     1.0    1.2     1.4     1.6     1.8     2.0X10-aSec. 


Fig.  17.     Oscillating-current  generator  charge. 


The  frequency  of  the  charging  oscillation  is 


=  71.2  cycles  per  sec.; 


for 


0.365  amp., 


(93) 


substituting  in  equations  (69)  and  (70)  we  have 

i=£-2ot  jo.365  cos  446.8  £  +  0.118  sin  446.8  t},  in  amp., 
and 
el  =  15,000  { 1  -£~20t  [cos  446.8  £-2.67  sin  446.8  £] } ,  in  volts, 

the  equations  of  condenser  charge. 

From  these  equations  the  values  of  i  and  el  are  plotted  in 
Fig.  18,  with  the  time  £  as  abscissas. 


As  seen,   the  value  i 


0.365  amp.,  is  reached  again 


at  the  time  t0  =  0.0012,  that  is,  after  30.6  time-degrees  or  about 
TV  of  a  period.  At  this  moment  the  condenser  e.m.f.  is  el  = 
e0  =  22,300  volts;  that  is,  by  setting  the  spark  gap  for  22,300 
volts  the  duration  of  the  condenser  charge  is  0.0012  second, 
or  in  other  words,  every  0.0012  second,  or  833  times  per  second, 
discharge  oscillations  are  produced. 

With  this  spark  gap,  the  charging  current  at  the  beginning 
and  at  the  end  of  the  condenser  charge  is  0.365  amp.,  and  the 


84 


TRANSIENT  PHENOMENA 


average  charging   current  is  0.3735  amp.  at  15,000  volts,  con- 
suming 5.6  kw. 

Assume  that  the  e.m.f.  at  the  condenser  terminals  at  the  end 
of  the  charge  is  e0  =  22,300  volts;  then  consider  two  cases, 
namely :  (a)  the  condenser  discharges  into  a  Tesla  transformer, 
and  (b)  the  condenser  discharges  into  an  iron  arc. 


•1 
20 
18 
16 
14 

§12 

5: 

3 

10 
8 

/ 

e  = 
r  .= 

=150 
=  100 

JO  vc 

noo 

Its 

/ 

* 

L  = 

C-= 

g. 

=  250 
=  0.0" 

h. 
_mJ 

/ 

=223 

10  vc 

Its 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

6 
4 
2 

/ 

/ 

( 

/ 

, 

/ 

4.0  ^ 


t  =  0.2          0.4          0.6          0.8         1.0         1.2  X  10'8Sec. 

Fig.  18.     Oscillating-current  generator  condenser  charge. 

(a)  The  Tesla  transformer,  that  is,  an  oscillating-current 
transformer,  has  no  iron,  but  a  primary  coil  of  very  few  turns 
(20)  and  a  secondary  coil  of  a  larger  number  of  turns  (360), 
both  immersed  in  oil. 

While  the  actual  ohmic  resistance  of  the  discharge  circuit  is 
only  0.1  ohm,  the  load  on  the  secondary  of  the  Tesla  trans- 
former, the  dissipation  of  energy  into  space  by  brush  discharge, 
etc.,  and  the  increase  of  resistance  by  unequal  current  distribu- 
tion in  the  conductor,  increase  the  effective  resistance  to  many 
times  the  ohmic  resistance.  We  can,  therefore,  assign  the 


OSCILLATING  CURRENTS 


85 


following  estimated  values:  r0=  20  ohms;  L0=  60  X  10~6  henry, 
and<7  =  2  X  10~8  farad. 
Then 


q0  =  108  ohms, 
=•0.898  X  10", 


2L 


-°  =  0.186, 

0.1667  X  106, 


sin  0.898 XlO6 1,  amp. 


•0  2   LQ 

which  give 

t=415  e~ 
and 

^  =  22,300  ,-o  1667x10^  {cos  o.898 X 106 1  +  0.186  sin  0.898 X 10'  Zj , 
volts. 

(94) 
The  frequency  of  oscillation  is 

0  SQS  v  1  f)6 

//_  r  ^T    =  143'000  cycles  per  seC;         (95) 

Fig.  19  shows  the  current  i  and  the  condenser  potential  el 
during  the  discharge,  with  the  time  t  as  abscissas.  As  seen, 
the  discharge  frequency  is  very  high  compared  with  the  fre- 


300 
25 
200 

1  5  a 

^ 

'o 

=»|20loHms 

( 

\ 

L 

-=,0.06  mh'. 

N 

1 

S'4 

C 

M|  if  i 

l^ 

\ 

6'i 

= 

22300 

vcilts 

1 

\l 

j 

\ 

\ 

< 

"^ 

s 

30^0 

-5 

± 

V 

\ 

/ 

^ 

^ 

•**^ 

^= 

^ 

•<. 

^ 

—  -1 

- 

* 

-15 

-200 

( 

<*. 

^^ 

\ 

/ 

. 

' 

1 

)        1.5         3        4.5        6         7.5         9       10.5      J2       13.5 

XlO'6    See. 


Fig.  19.     Oscillating-current  generator  condenser  discharge. 

quency  of  charge,  the  duration  of  discharge  very  short,  and 
the  damping  very  great;  a  decrement  of  0.55,  so  that  the  oscil- 
lation dies  out  very  rapidly.  The  oscillating  current,  however, 
is  enormous  compared  with  the  charging  current;  with  a  mean 
charging  current  of  0.3735  amp.,  and  a  maximum  charging 
current  of  0.378  amp.  the  maximum  discharge  current  is  315 
amp.,or  813  times  as  large  as  the  charging  current. 


86  TRANSIENT  PHENOMENA 

The  effective  value  of  the  discharge  current,  from  equation 
(87),  is  il  =  14.4  amp.,  or  nearly  40  times  the  charging  current. 

53.  (b)  When  discharging  the  condenser  directly,  through 
an  ultraviolet  or  iron  arc,  in  a  straight  path,  and  estimating 
r0  =  5  ohms  and  L0  =  4  X  10~6  henry,  we  have 

g0  =  27.84  ohms,  ^  -  0.1795,. 

#0 

£-  =  3.48  X  10°,  ^f-  =  0.625  X  10"; 

2  LQ  2  L0 

then, 

t  =  1600r-0625xl°6'  sin  3.48  X  106£,  in  amp., 
and 

^  =  22,300  £- 0625X1°6<  { cos  3.48 X 106  ^  +  0.1795  sin  3.48 X 106  ^j 
in  volts, 

(96) 
and  the  frequency  of  oscillation  is 

/0  -  562,000  cycles  per  sec.;  (97) 

that  is,  the  frequency  is  still  higher,  over  half  a  million 
cycles;  the  maximum  discharge  current  over  1000  amperes; 
however,  the  duration  of  the  discharge  is  still  shorter,  the 
oscillations  dying  out  more  rapidly. 

The  effective  value  of  the  discharge  current,  from  (87),  is 
tj  =  28.88  amp.,  or  77  times  the  charging  current.  A  hot 
wire  ammeter  in  the  discharge  circuit  in  this  case  showed 
29  amp. 

As  seen,  with  a  very  small  current  supply,  of  0.3735  amp., 
at  e  =  15,000  volts,  in  the  discharge  circuit  a  maximum  voltage 
of  22,300,  or  nearly  50  per  cent  higher  than  the  impressed 
voltage,  is  found,  and  a  very  large  current,  of  an  effective  value 
very  many  times  larger  than  the  supply  current. 

As  a  rule,  instead  of  a  constant  impressed  e.m.f.,  e,  a  low 
frequency  alternating  e.m.f.  is  used,  since  it  is  more  conven- 
iently generated  by  a  step-up  transformer.  In  this  case  the 
condenser  discharges  occur  not  at  constant  intervals  of  t0  sec- 
onds, but  only  during  that  part  of  each  half  wave  when  the 
e.m.f.  is  sufficient  to  jump  the  gap  eot  and  at  intervals  which 
are  shorter  at  the  maximum  of  the  e.m.f.  wave  than  at  its 
beginning  and  end. 


OSCILLATING  CURRENTS  87 

For  instance,  using  a  step-up  transformer  giving  17,400  volts 
effective  (by  the  ratio  of  turns  1  :  150,  with  118  volts  im- 
pressed at  60  cycles),  or  a  maximum  of  24,700  volts,  then 
during  each  half  wave  the  first  discharge  occurs  as  soon  as  the 
voltage  has  reached  22,300,  sufficient  to  jump  the  spark  gap, 
and  then  a  series  of  discharges  occurs,  at  intervals  decreasing 
with  the  increase  of  the  impressed  e.m.f.,  up  to  its  maximum, 
and  then  with  increasing  intervals,  until  on  the  decreasing 
wave  the  e.m.f.  has  fallen  below  that  which,  during  the  charg- 
ing oscillation,  can  jump  the  gap  e0,  that  is,  about  13,000  volts. 
Then  the  oscillating  discharges  stop,  and  start  again  during  the 
next  half  wave. 

Hence  the  phenomenon  is  of  the  same  character  as  investi- 
gated above  for  constant  impressed  e.m.f.,  except  that  it  is 
intermittent,  with  gaps  during  the  zero  period  of  impressed 
voltage  and  unequal  time  intervals  tQ  between  the  successive 
discharges. 

54.  An  underground  cable  system  can  act  as  an  oscillating- 
current  generator,  with  the  capacity  of  the  cables  as  condenser, 
the  internal  inductance  of  the  generators  as  reactor,  and  a  short- 
circuiting  arc  as  discharge  circuit. 

In  a  cable  system  where  this  phenomenon  was  observed 
the  constants  were  approximately  as  follows:  capacity  of  the 
cable  system,  C  =  102  mf.;  inductance  of  30,000-kw.  in  gen- 
erators, L  =  6.4  mh.;  resistance  of  generators  and  circuit  up  to 
the  short-circuiting  arc,  r  =  0.1  ohm  and  r  =  1.0  ohm  respec- 
tively; impressed  e.m.f.,  11,000  volts  effective,  and  the  fre- 
quency 25  cycles  per  second. 

The  frequency  of  charging  oscillation  in  this  case  is 

/  =  ~=  =  197  cycles  per  sec. 

4  TTL/ 

since 

q  =  \  —  -  r2  =  15.8  ohms. 
C 

Substituting  these  values  in  the  preceding  equations,  and 
estimating  the  constants  of  the  discharge  circuit,  gives  enor- 
mous values  of  discharge  current  and  e.m.f. 


CHAPTER  VII. 

RESISTANCE,   INDUCTANCE,  AND  CAPACITY  IN  SERIES  IN 
ALTERNATING-CURRENT  CIRCUIT. 

65.  Let,  at  time  t  =  0  or  0  =  0,  the  e.m.f., 

e  =  E  cos  (0  -  00),  (1) 

be  impressed  upon  a  circuit  containing  in  series  the  resistance,  r, 
the  inductance,  L,  and  the  capacity,  C. 

The  inductive  reactance  is        x  =  2  TT/L     1 
and  the  condensive  reactance  is  xc  =  > 

2  7T/C          J 

where/  =  frequency  and  6  =  2  nft.  (3) 

Then  the  e.m.f.  consumed  by  resistance  is  ri\ 
the  e.m.f.  consumed  by  inductance,  is 

di         di 

Ldt  =  xJe' 

and  the  e.m.f.  consumed  by  capacity  is 

,  (4) 


where  i  =  instantaneous  value  of  the  current. 

di 


di  C 

Hence,  e  =  ri  +  x  --  +  xc  I  i  dO,  (5) 

da          J 

di  f* 

E  cos  (6  -  00)  =  ri  +  x  —  +  xc  li  dO,  (6) 

da          J 


and  hence,  the  difference. of  potential  at  the  condenser  terminals 
is 

el  =  xc  I  idd  =  E  cos  (6  -  00)  -  ri  -  x  —  -  (7) 

u  cw 

88 


RESISTANCE,   INDUCTANCE,   AND  CAPACITY  89 

Equation  (6)  differentiated  gives 

E  sin  (0  -  0Q)  +  x-jj^  +  r  -^  +  xei  =  0.  (8) 

The  integral  of  this  equation  (8)  is  of  the  general  form 

i  =  Ara9  +  B  cos  (6  -  o-).  (9) 

Substituting  (9)  in  (8),  and  rearranging,  gives 
A£~ae\a2x— ar  +  xr\  +  sin#  \E  cos  On—rB  cosv—B  (x—xc)  sin  o-j 

(  tj'(  y  \c,/j 

-  cos  0  {£"  sin  6Q  —  rB  sin  a-  +  J5  (x  —  xc)  cos  <r}  =  0, 

and,  since  this  must  be  an  identity, 

a?x  —  ar  +  xc  =  0, 
E  cos  #0  —  rB  cos  a-  —  B  (x  —  xc)  sin  o-  =  0,      ,      (10) 

#  sin  00  --  rB  sin  o-  +  B  (x  —  xc)  cos  a-  =  0. 


Substituting 


—  4 


tan  7  = 


in  equations  (10)  gives 


x  -  xc 


r  ±  s 


and 


=  00  +  ? 

=  indefinite, 


and  the  equation  of  current,  (9),  thus  is 
i  =  -cos  (6  -  60  -7)  +  A/~ 


(ID 


(12) 


(13) 


90  TRANSIENT  PHENOMENA 

and,  substituting   (12)   in   (7),  and  rearranging,  the  potential 
difference  at  the  condenser  terminals  is 


.    (14) 


The  two  integration  constants  Al  and  A2  are  given  by  the 
terminal  conditions  of  the  problem. 
Let,  at  the  moment  of  start, 


0-0, 

i  =  iQ  =  instantaneous  value  of  current  and 

e1  =  e0  =  instantaneous  value  of  condenser  potential 
difference. 

Substituting  in  (13)  and  (14), 

ET 


(15) 


(#o  +  7)  +  A,  +  A, 
zn 


and 


•  Q 

Therefore 


Aj  +  Aj^^-  -cos  (60  +  7) 

and 

n0  +  2e0      E  , 
Aj-A2=-  -  +  --  rcos(<90  +  7)-2^csm(i9o+7)   , 

S  SZ0 


(16) 


or, 

r—  s  . 


'  e 


and 


S  SZ 

r  +  s  . 

2   %^"eo       -fi 
-  -cos 


(17) 


RESISTANCE,   INDUCTANCE,   AND  CAPACITY 


91 


Substituting  (17)  in  (13)  and  (14)  gives  the  integral  equations 
of  the  problem. 
The  current  is 


Tfl 


Tjl     f  _   T~S  Q    p 


p 

^ 

\_    J 


and  the  potential  difference  at  the  condenser  terminals  is 


E 


where 


and 


r 

r-±£ 


Cos 


r+s 


(x  - 


x  —  xc 
tan  7 .  =  -     —  > 


=  Vr2  -  4  x  xr 


(19) 


(ID 


The  expressions  of  i  and  et  consist  of  three  terms  each : 

(1)  The  permanent  term,  which  is  the  only  one  remaining 
after  some  time; 

(2)  A  transient  term  depending  upon  the  constants  of  the 
circuit,  r,  s,  xci  z0,  x,  the  impressed  e.m.f.,  E,  and  its  phase  00  at 
the  moment  of  starting,   but  independent  of  the  conditions 
existing  in  the  circuit  before  the  start;  and 


92  TRANSIENT  PHENOMENA 

(3)  A  term  depending,  besides  upon  the  constants  of  the 
circuit,  upon  the  instantaneous  values  of  current  and  potential 
difference,  iQ  and  e0,  at  the  moment  of  starting  the  circuit,  and 
thereby  upon  the  electrical  conditions  of  the  circuit  before 
impressing  the  e.m.f.,  e.  This  term  disappears  if  the  circuit  is 
dead  before  the  start. 


Equations  (18)  and  (19)  contain  the  term     s  =  Vr2  —  4  x  xc 
=  V/7*2  —  4  —  ;  hence  apply  only  when    r2  >  4  x  xc,  but  become 

indeterminate  if.  r2=4xxc}  and  imaginary  if  r2<  4  x  xc;  in  the 
latter  cases  they  have  to  be  rearranged  so  as  to  appear  in  real 
form,  in  manner  similar  to  that  in  Chapter  V. 

56.   In  the  critical  case,  r2  =  4  xxc  and  s  =  0,  equation  (18), 
rearranged,  assumes  the  form 

TJ1  Tfl H  £ 

i  =  -  cos  (0  -  00-7)  +  -£ 

-cos(00+7)-zcsin>(00  +  fy)   -  -—  cos(00+7)[ 

_2  s  ) 


However,  developing  in  a  series,  and  canceling  all  but  the 
first  term  as  infinitely  small,  we  have 


- 
e    2x    -e 


-  > 

X 


hence  the  current  is 

EJ  ^      _JL 

2x 


(  I-f  ~\  ft  ) 

\  \-  cos  (00+  7)  -  xc  sin  (00  +  7)   -    -  cos  (00  +  7)  ( 
(\_2  Jx 


(20) 


RESISTANCE,   INDUCTANCE,   AND  CAPACITY 


93 


and   in  the  same  manner  the  potential  difference  at  condenser 
terminals  is 


Ex, 


E    -™ 


[T^  ~l  fi 

-  cos  (00  +  7)  -  xcr  sin  (00  +  7)  I  -  -  2  xc  sin  (0  +  7) 


(21) 


Here  again  three  terms  exist,  namely:  a  permanent  term,  a 
transient  term  depending  only  on  E  and  00,  and  a  transient 
term  depending  on  iQ  and  e0. 

57.  In  the  trigonometric  or  oscillatory  case,  r2  <  4  a;  xc,  s  be- 
comes imaginary,  and  equations  (18)  and  (19)  therefore  contain 
complex  imaginary  exponents,  which  have  to  be  eliminated, 
since  the  complex  imaginary  form  of  the  equation  obviously 
is  only  apparent,  the  phenomenon  being  real. 

Substituting 

q  =  V4  x  xc  -  r2  =  js  (22) 

in  equations  (13)  and  (14),  and  also  substituting  the  trigono- 
metric expressions 


and 


and  separating  the  imaginary  and  the  real  terms,  gives 


(23) 


E 


(A,  4-  A2)  cos  -^ 


-  —9 

: 


j  (A,  -  A,)  sin        6 


94  TRANSIENT  PHENOMENA 

and 


then  substituting  herein  the  equations  (16)  and  (22)  the  imagi 
nary  disappears,  and  we  have  the  current, 


-0 
2x 


JL  a      [2xc  • 

!e°^K°sin2lei'  (24) 

and  the  potential  difference  at  the  condenser  terminals, 


Here  the  three  component  terms  are  seen  also. 

58.  As  examples  are  shown  in  Figs.  20  and  21,  the  starting 
of  the  current  i,  its  permanent  term  i1  ',  and  the  two  transient 
terms  i1  and  iv  and  their  difference,  for  the  constants  E  =  1000 
volts  =  maximum  value  of  impressed  e.m.f.;  r  =  200  ohms 
=  resistance  ;  x  =  75  ohms  =  inductive  reactance,  and  xc  =  75 
ohms  =  condensive  reactance.  We  have 

4  x  xc  =  22,500 

and  r2  =  40,000; 

therefore 

r2  >  4  x  xc, 


RESISTANCE,   INDUCTANCE,  AND  CAPACITY 


95 


that  is,  the  start  is  logarithmic,  and  z0  =  200,  s  =  132,  and 
7  =  0. 


20 


60        80       100       120       140       160      180      200 
Degrees 


Fig.  20.     Starting  of  an  alternating-current  circuit,  having  capacity,  inductance 
and  resistance  in  series.     Logarithmic  start. 

In  Fig.  20  the  circuit  is  closed  at  the  moment  00  =  0,  that 
is,  at  the  maximum  value  of  the  impressed  e.m.f.,  giving  from 
the  equations  (18)  and  (19),  since  i0  =  0,  e0  =  0, 


and 


i  =  5  {cos  6  -  1.26  s-2-22'  +  0.26  £-°'452' } 
el  =  375  {sin0  +  0.57  (e-»-»«_fi-o.462*)}p 


0        20        40 


100      120       140       160       180       200 
Degrees 


Fig.  21.     Starting  of  an  alternating-current  circuit  having  capacity,  inductance 
and  resistance  in  series.     Logarithmic  start. 

In  Fig.  21  the  circuit  is  closed  at  the  moment  00  =  90°,  that 
is,  at  the  zero  value  of  the  impressed  e.m.f.,  giving  the  equa- 
tions 

i  =  5  {sinfl  +  0.57  Or2'22'  -  fi-o-«")} 
and 

e,  =  -  375  {cosfl  +  0.26  *-'•»•-  1.26  e~°-™°)}. 


96 


TRANSIENT  PHENOMENA 


There  exists  no  value  of   #0  which  does  not  give  rise  to  a 
transient  term. 


-2 


-4 


E  =1000  tolts 
200-phiii 
133.3  otii 


20 


80       100       120      140 


180      200 


Fig.  22.     Starting  of  an  alternating-current  circuit  having  capacity,  inductance 
and  resistance  in  series.     Critical  start. 

In  Fig.  22  the  start  of  a  circuit  is  shown,  with  the  inductive 
reactance  increased  so  as  to  give  the  critical  condition, 

r2  =  4zzc, 

but  otherwise  the  constants  are  the  same  as  in  Figs.  20  and  21, 
that  is,  E  =  1000  volts;  r  =  200  ohms;  x  =  133.3  ohms,  and 
xc  =  75  ohms; 

therefore          z0  =  208.3, 

fro  o 

tan  7  =  ^   -  =  0.2915,    or    7  =  16°, 
ZOO 

assuming  that  the  circuit  is  started  at  the  moment  00  =  0,  or 
at  the  maximum  value  of  impressed  e.m.f. 
Then  (20)  and  (21)  give 

i  =  4.78  cos  (6  -  16°)  +  r°'75'  (2.7  0  -  4.6) 
and 

e,=  358 sin  (6  -  16°)  -  £-°-75'(410#  -  99). 

Here  also  no  value  of  00  exists  at  which  the  transient  term 
disappears. 

69.  The  most  important  is  the  oscillating  case,  r2  <  4  x  xc, 
since  it  is  the  most  common  in  electrical  circuits,  as  underground 
cable  systems  and  overhead  high  potential  circuits,  and  also  is 
practically  the  only  one  in  which  excessive  currents  or  excessive 
voltages,  and  thereby  dangerous  phenomena,  may  occur. 


RESISTANCE,   INDUCTANCE,   AND  CAPACITY 


97 


If  the  condensive  reactance  xc  is  high  compared  with  the 
resistance  r  and  the  inductive  reactance  x,  the  equations  of 
start  for  the  circuit  from  dead  condition,  that  is,  t'0  =  0  and 
e0  =  0,  are  found  by  substitution  into  the  general  equations 
(24)  and  (25),  which  give  the  current  as 


E(   .    , 

{  sm  (0  — 

Xc  f 


2xTsin00cosV/-c0 
L  r  x 


(26) 


and  the  potential  difference  at  the  condenser  terminals  as 


cos#  cos  V  -H 
where 


cos 


xc 


sin  ,     (27) 


xc,  and  7  =  -  90°. 


(28) 


In  this  case  an  oscillating  term  always  exists  whatever  the 
value  of  00,  that  is,  the  point  of  the  wave,  where  the  circuit  is 
started. 

The  frequency  of  oscillation  therefore  is 


/o 

or,  approximately, 


2x" 


_ 
4X2 


(29) 


where/  =  fundamental  frequency. 

Substituting  x  =  2nfL  and  zc  =  — -r-,  we  have 


CL 


or,  approximately, 

/o 


(30) 


98  TRANSIENT  PHENOMENA 

60.  The  oscillating  start,  or,  in  general,  change  of  circuit 
conditions,  is  the  most  important,  since  in  circuits  containing 
capacity  the  transient  effect  is  almost  always  oscillating. 

The  most  common  examples  of  capacity  are  distributed 
capacity  in  transmission  lines,  cables,  etc.,  and  capacity  in  the 
form  of  electrostatic  condensers  for  neutralizing  lagging  currents, 
for  constant  potential-constant  current  transformation,  etc. 

(a)  In  transmission  lines  or  cables  the  charging  current  is  a 
fraction  of  full-load  current  i0,  and  the  e.m.f.  of  self-inductance 
consumed  by  the  line  reactance  is  a  fraction  of  the  impressed 
e.m.f.  e0.  Since,  however,  the  charging  current  is  (approximately) 

p 

=  —  and  the  e.m.f.  of  self-inductance  =  xi0,  we  have 

Xc 


eo 

<  i0,  xi  0 

Xc 


hence,  multiplying, 

x 

—  <  1  and  x  <  xc. 

Xc 

The  resistance  r  is  of  the  same  magnitude  as  x\  thus 

4xxc  >r\ 

For  instance,  with  10  per  cent  resistance  drop,  30  per  cent 
reactance  voltage,  and  20  per  cent  charging  current  in  the  line, 
assuming  half  the  resistance  and  reactance  as  in  series  with  the 
capacity  (that  is,  representing  the  distributed  capacity  of  the 
line  by  one  condenser  shunted  across  its  center)  and  denoting 

?-*• 

where  e0  =  impressed  voltage,  i0  =  full-load  current,  we  have 


x  =  0.5  X  0.3  p  =  0.15  p, 

r  =  0.5  X  0.1  p  =  0.05  p, 
and 

r  -5-  s  -s-  zc  =  1  -f-  3  -r-  100, 
and 

4  x  xr  +  r2  =  1200  -4-  1. 


RESISTANCE,  INDUCTANCE,  AND  CAPACITY  99 

In  this  case,  to  make  the  start  non-oscillating,  we  must  have 

x  <  — — -  r,  or  x  <  0.000125  p,  which  is  not  possible;  t>r  r  > 

which  can  be  done  only  by  starting  the  circuit  through  a  very 
large  non-inductive  resistance  (of  such  size  as  to  cut  the  starting 

current  down  to  less  than  —  of  full-load  current).     Even  in 

this  case,  however,  oscillations  would  appear  by  a  change  of 
load,  etc.,  after  the  start  of  the  circuit. 

(6)  When  using  electrostatic  condensers  for  producing  watt- 
less leading  currents,  the  resistance  in  series  with  the  condensers 
is  made  as  low  as  possible,  for  reasons  of  efficiency. 

Even  with  the  extreme  value  of  10  per  cent  resistance,  or 

r  4-  xc  =  I  -f-  10,  the  non-oscillating  condition  is  x  <  —  r,  or 

0.23  per  cent,  which  is  not  feasible. 
In  general,  if 

x  consumes 12  4  9  16  per  cent  of  the  con- 
denser potential 
difference, 

r  must  consume  >  20  28.3  40  60  80  per  cent  of  the  con- 
denser potential 
difference. 

That  is,  a  very  high  non-inductive  resistance  is  required  to 
avoid  oscillations. 

The    frequency    of   oscillation    is   approximately  /0  =  y  —  / 

that  is,  is  lower  than  the  impressed  frequency  if  xc  <  x  (or  the 
permanent  current  lags),  and  higher  than  the  impressed  fre- 
quency if  xc  >  x  (or  the  permanent  current  leads).  In  trans- 
mission lines  and  cables  the  latter  is  always  the  case. 

Since  in  a  transmission  line— is  approximately  the  charging 

Xc 

X 

current,  as  fraction   of  full-load  current,  and  ~-  half  the  line 

e.m.f.  of  self-inductance,  or  reactance  voltage,  as  fraction  of 
impressed  voltage,  the  following  is  approximately  true : 


100 


TRANSIENT  PHENOMENA 


The  frequency  of  oscillation  of  a  transmission  line  is  the 
impressed  frequency  divided  by  the  square  root  of  the  product 
of  charging  current  and  of  half  the  reactance  voltage  of  the  line, 
given  respectively  as  fractions  of  full-load  current  and  of  im- 
pressed voltage.  For  instance,  10  per  cent  charging  current, 
20  per  cent  reactance  voltage,  gives  an  oscillation  frequency 


vui  x  o.i 


10  f. 


Fig.  23.     Starting  of  an  alternating-current  circuit  having  capacity,  inductance 
and  resistance  in  series.     Oscillating  start  of  transmission  line. 

61.  In  Figs.  23  and  24  is  given  as  example  the  start 
of  current  in  a  circuit  having  the  constants,  E  =  35,000 
cos  (6  —  00);  r  =  5  ohms;  x  =  10  ohms,  and  xc  =  1000  ohms. 

In  Fig.  23  for  00=  0°,  or  approximately  maximum  oscilla- 
tion, 

i  =  -  35  {sin  0  -  10  e-  °25 '  sin  10  6} 
and 

el  =  35,000  {cos  6  -  e~  °25  •  [cos  10  0  +  0.025  sin  10  0]} . 

In  Fig.  24  for  #0  =  90°,  or  approximately  minimum  oscilla- 
tion, 

i  =  35  {cos  0  -  r  °25  9  cos  10  6}  \ 

and 

e,  =  35,000  { sin  0  +  0.1  <r  ° 25  °  sin  10  0 } . ) 

As  seen,  the  frequency  is  10  times  the  fundamental,  and  in 
starting  the  potential  difference  nearly  doubles. 


RESISTANCE,  INDUCTANCE,  AND  CAPACITY 


101 


As  further  example,  Fig.  25  shows  the  start  of  a  circuit  of  a 
frequency  of  oscillation  of  the  same  magnitude  as  the  funda- 
mental, in  resonance  condition,  x  =  xc,  and  of  high  resistance. 


60 
40 
20 
0 

-20 

-40 

-60 

60 

40 

20 

0 

-20 
-40 


\s 


4 


\ 


=  35000  volts 
-5-ohW- 
10  oh' ma1 


-\ 


100  ohms 

gd-p- 


Fig.  24.     Starting  of  an  alternating-current  circuit  having  capacity,  inductance 
and  resistance  in  series.     Oscillating  start  of  transmission  line. 


The  circuit  constants  are  E  =  1500  volts;  r  =  30  ohms; 
x  =  20  ohms;  xc  =  20  ohms,  and  00  =  —  7;  which  give 
3  =  26.46;  z0  =  30;  7  =  0,  and  00  =  0. 


Fig.  25.     Starting  of  an  alternating-current  circuit  having  capacity,  inductance 
and  resistance  in  series.     Oscillating  start.     High  resistance. 

Substituting  in  equations  (24)  and  (25)  gives 
i  =  50  {cos  0  -  r  °-75  0  [cos  0.661  Q  -  1.14  sin  0.661  6]}  ] 

=  1000  {sin  e  -  1.51  e- ° 75  •  sin  0.661  6}. 


102 


TRANSIENT  PHENOMENA 


As  example  of  an  oscillation  of  long  wave,  Fig.  26  represents 
the  start  of  a  circuit  having  the  constants  E  =  1500  volts; 
r  =  10  ohms;  x  =  62.5  ohms;  xc  =  10  ohms,  and  00  =  —  7; 
which  give  q  =  49;  20  =  53.4;  7  =  79°,  and  00  =  -  79°. 

Substituting  in  equations  (24)  and  (25)  gives 


i  =  28  f  cos  0  -  e 


and 


[cos  0.390  -  0.2  sin  0.390]} 


e,  =  280  {sin  6  -  2.55  r  °-°8 «  sin  0.396  0 


62.  While  in  the  preceding  examples,  Figs.  23  to  26,  con- 
stants of  transmission  lines  have  been  used,  as  will  be  shown 
in  the  following  chapters,  in  the  case  of  a  transmission  line 


V 


9 

xt. 


62.5  ohms 


Fig.  26.     Starting  of  an  alternating-current  circuit  having  capacity,  inductance 
and  resistance  in  series.     Oscillating  start  of  long  period. 

with  distributed  capacity  and  inductance,  the  oscillation  does 
not  consist  of  one  definite  frequency  but  an  infinite  series  of 
frequencies,  and  the  preceding  discussion  thus  approximates 
only  the  fundamental  frequency  of  the  system.  This,  however, 
is  the  frequency  which  usually  predominates  in  a  high  power 
low  frequency  surge  of  the  system. 

In  an  underground  cable  system  the  preceding  discussion 
applies  more  closely,  since  in  such  a  system  capacity  and  induc- 
tance are  more  nearly  localized :  the  capacity  is  in  the  under- 
ground cables,  which  are  of  low  inductance,  and  the  inductance 
is  in  the  generating  system,  which  has  practically  no  capacity. 

In  an  underground  cable  system  the  tendency  therefore  is 


RESISTANCE,  INDUCTANCE,  AND  CAPACITY  103 

either  towards  a  local,  very  high  frequency  oscillation,  or  travel- 
ing wave,  of  very  limited  power,  in  a  part  of  the  cables,  or  a  low 
frequency  high  power  surge,  frequently  of  destructive  magnitude, 
of  the  joint  capacity  of  the  cables,  against  the  inductance  of  the 
generating  system. 

63.  The  physical  meaning  of  the  transient  terms  can  best  be 
understood  by  reviewing  their  origin. 

In  a  circuit  containing  resistance  and  inductance  only,  but  a 
single  transient  term  appears  of  exponential  nature.  In  such  a 
circuit  at  any  moment,  and  thus  at  the  moment  of  start,  the 
current  should  have  a  certain  definite  value,  depending  on 
the  constants  of  the  circuit.  In  the  moment  of  start,  however, the 
current  may  have  a  different  value,  depending  on  the  preceding 
condition,  as  for  instance  the  value  zero  if  the  circuit  has  been 
open  before.  The  current  thus  adjusts  itself  from  the  initial 
value  to  the  permanent  value  on  an  exponential  curve,  which 
disappears  if  the  initial  value  happens  to  coincide  with  the  final 
value,  as  for  instance  if  the  circuit  is  closed  at  the  moment  of 
the  e.m.f.  wave,  when  the  permanent  current  should  be  zero. 
The  approach  of  current  to  the  permanent  value  is  retarded  by 
the  inductance,  accelerated  by  the  resistance  of  the  circuit. 

In  a  circuit  containing  inductance  and  capacity,  at  any 
moment  the  current  has  a  certain  value  and  the  condenser  a 
certain  charge « that  is,  potential  difference.  In  the  moment  of 
start,  current  intensity  and  condenser  charge  have  definite 
values,  depending  on  the  previous  condition,  as  zero,  if  the 
circuit  was  open,  and  thus  two  transient  terms  must  appear, 
depending  upon  the  adjustment  of  current  and  of  condenser 
e.m.f.  to  their  permanent  values. 

Since  at  the  moment  when  the  current  is  zero  the  condenser 
e.m.f.  is  maximum,  and  inversely,  in  a  circuit  containing  induc- 
tance and  capacity,  a  change  of  circuit  conditions  always  results 
in  the  appearance  of  a  transient  term. 

If  the  circuit  is  closed  at  the  moment  when  the  condenser 
e.m.f.  should  be  zero,  that  is,  about  the  maximum  value  of  cur- 
rent, the  transient  term  of  current  cannot  exceed  in  amplitude  its 
final  value,  since  its  maximum  or  initial  value  equals  the  value 
which  the  current  should  have  at  this  moment.  If,  however, 
the  circuit  is  closed  at  the  moment  where  the  current  should  be 
zero  and  the  condenser  e.m.f.  maximum,  the  condenser  being 


104  TRANSIENT  PHENOMENA 

without  charge  acts  in  the  first  moment  like  a  short  circuit,  that 
is,  the  current  begins  at  a  value  corresponding  to  the  impressed 
e.m.f.  divided  by  the  line  impedance.  Thus  if  we  neglect  the 
resistance  and  if  the  condenser  reactance  equals  n2  times  line 
reactance,  the  current  starts  at  n2  times  its  final  rate;  thus  it 
would,  in  a  half  wave,  give  n2  times  the  full  charge  of  the  con- 
denser, or  in  other  words,  charge  the  condenser  in  -  of  the  time 
of  a  half  wave.  That  is,  the  period  of  the  starting  current  is 

-  and  the  amplitude  n  times  that  of  the  final  current.  How- 
n  l 

ever,  as  soon  as  the  condenser  is  charged,  in  -  of  a  period  of 

Ti 

the  impressed  e.m.f.,  the  magnetic  field  of  the  charging  current 
produces  a  return  current,  discharging  the  condenser  again  at 
the  same  rate. 

Thus  the  normal  condition  of  start  is  an  oscillation  of  such  a 
frequency  as  to  give  the  full  condenser  charge  at  a  rate  which 
when  continued  up  to  full  frequency  would  give  an  amplitude 
equal  to  the  impressed  e.m.f.  divided  by  the  line  reactance. 
The  effect  of  the  line  resistance  is  to  consume  e.m.f.  and  thus 
dampen  the  oscillation,  until  the  resistance  consumes  during  the 
condenser  charge  as  much  energy  as  the  magnetic  field  would 
store  up,  and  then  the  oscillation  disappears  and  the  start  becomes 
exponential. 

Analytically  the  double  transient  term  appears  as  the  result 
of  the  two  roots  of  a  quadratic  equation,  as  seen  above. 


CHAPTER  VIII. 

LOW  FREQUENCY  SURGES  IN  HIGH  POTENTIAL  SYSTEMS. 

64.  In  electric  circuits  of  considerable  capacity,  that  is,  in 
extended  high  potential  systems,  as  long  distance  transmission 
lines  and  underground  cable  systems,  occasionally  destructive 
high  potential  low  frequency  surges  occur;  that  is,  oscillations 
of  the  whole  system,  of  the  same  character  as  in  the  case  of 
localized  capacity  and  inductance  discussed  in  the  preceding 
chapter. 

While  a  system  of  distributed  capacity  has  an  infinite  number 
of  frequencies,  which  usually  are  the  odd  multiples  of  a  funda- 
mental frequency  of  oscillation,  in  those  cases  where  the 
fundamental  frequency  predominates  and  the  effect  of  the 
higher  frequencies  is  negligible,  the  oscillation  can  be  approxi- 
mated by  the  equations  of  oscillation  given  in  Chapters  V  and 
VII,  which  are  far  simpler  than  the  equations  of  an  oscillation 
of  a  system  of  distributed  capacity. 

Such  low  frequency  surges  comprise  the  total  system,  not  only 
the  transmission  lines  but  also  the  step-up  transformers,  gen- 
erators, etc.,  and  in  an  underground  cable  system  in  such  an 
oscillation  the  capacity  and  inductance  are  indeed  localized  to 
a  certain  extent,  the  one  in  the  cables,  the  other  in  the  generating 
system.  In  an  underground  cable  system,  therefore,  of  the 
infinite  series  of  frequencies  of  oscillations  which  theoretically 
exist,  only  the  fundamental  frequency  and  those  very  high 
harmonics  which  represent  local  oscillations  of  sections  of 
cables  can  be  pronounced,  and  the  first  higher  harmonics  of  the 
fundamental  frequency  must  be  practically  absent.  That  is, 
oscillations  of  an  underground  cable  system  are  either 

(a)  Low  frequency  high  power  surges  of  the  whole  system, 
of  a  frequency  of  a  few  hundred  cycles,  frequently  of  destructive 
character,  or, 

(6)  Very  high  frequency  low  power  oscillations,  local  in 
character,  so  called  "static,"  probably  of  frequencies  of  hundred 

105 


106  TRANSIENT  PHENOMENA 

thousands  of  cycles,  rarely  directly  destructive,  but  indirectly 
harmful  in  their  weakening  action  on  the  insulation  and  the 
possibility  of  their  starting  a  low  frequency  surge. 

The  former  ones  only  are  considered  in  the  present  chapter. 
Their  causes  may  be  manifold,  —  changes  of  circuit  conditions,  as 
starting,  opening  a  short  circuit,  existence  of  a  flaring  arc  on  the 
system,  etc. 

In  the  circuit  from  the  generating  system  to  the  capacity  of 
the  transmission  line  or  the  underground  cables,  we  have  always 

r2  <  —j-;  that  is,  the  phenomenon  is  always  oscillatory,  and 
(_/ 

equations  (24)  and  (25),  Chapter  VII,  apply,  and  for  the  current 
we  have 


~n> 


(1) 


and  for  the  condenser  potential  we  have 
c 


Z      X 


(2) 

65.  These  equations  (1)  and  (2)  can  be  essentially  simplified 
by  neglecting  terms  of  secondary  magnitude. 

xc  is  in  high  potential  transmission  lines  or  cables  always  very 
large  compared  with  r  and  x. 

The  full-load  resistance  and  reactance  voltage  may  vary 
from  less  than  5  per  cent  to  about  20  per  cent  of  the  impressed 
e.m.f.,  the  charging  current  of  the  line  from  5  per  cent  to 
about  20  per  cent  of  full-load  current,  at  normal  voltage  and 
frequency. 

In  this  case,  xc  is  from  25  to  more  than  400  times  as  large  as  r 
or  x,  and  r  and  x  thus  negligible  compared  with  xc. 


HIGH  POTENTIAL  SYSTEMS 
It  is  then,  in  close  approximation : 


q  =  2  Vx  xc 


= -      = -90o. 


107 


(3) 


Substituting  these  values  in  equations  (1)  and  (2)  gives  the 
current  as 

i=  --  sin(0-#0)  +  e~2^  \\\  --  sin  00~|  cosy  -  0 
xc  (  |_        xc         J  x 


and  the  potential  difference  at  the  condenser  as 
l  =  E  cos  (0  -  00)  +  [e0  -  J^  cos  0J  cos 


T2re0  +  4xa:ct0  .    _ 
4VxZ  4 


E 


XX, 


in  00) Jsin  y/^-c 


(2  r  cos  00  +  4  x  sin 
These  equations  consist  of  three  terms: 

x,        .  _     x,    /        I        p    "     -I-     £    "'  • 

el  ~   61         Kl  1     ;   . 

^     '       (a          ft  \ 

I  — Sin       (C7       —        UQ), 


(5) 


(6) 


108 


TRANSIENT  PHENOMENA 


E    -~e  ( 
= £    2*    ) 


cos  6, 


cos  00  cos  V  -  0  +         ''cos  6> 


-  Ee 


or,  by  dropping  terms  of  secondary  order, 
E      -  ~e 


—  Ee    2x   cos  6 n  cos  V/  — 


and: 


or,  by  dropping  terms  of  secondary  order, 


Vx 


cos 

x 


(7) 


(8) 


(9) 


(10) 


Thus  the  £ota£  current  is  approximately 


e0  —  E  cos  60   . 


and  the  difference  of  potential  at  the  condenser  is 

—    r    9   C  I 

et=E  cos  (6  -  00)  +  £         j  (e0  -  £  cos  «„)  cos  \/ ^ 


(11) 


HIGH  POTENTIAL  SYSTEMS  109 

Of  the  three  terms:  i*  ',  e/;  in  \  e/';  i"r  ,  e/",  the  first  obviously 
represents  the  stationary  condition  of  charging  current  and  con- 
denser potential,  since  the  two  other  terms  disappear  for  t  =  oo  . 

The  second  term,  i",  e^',  represents  that  component  of  oscilla- 
tion which  depends  upon  the  phase  of  impressed  e.m.f.,  or  the 
point  of  the  impressed  e.m.f.  wave,  at  which  the  oscillation 
begins,  while  the  third  term,  inf  ,  e/",  represents  the  component 
of  oscillation  which  depends  upon  the  instantaneous  values  of 
current  and  e.m.f.  respectively,  at  the  moment  at  which  the 

oscillation  begins,   s      c  is  the  decrement  of  the  oscillation. 
66.   The  frequency  of  oscillation  is 


where  /  is  the  impressed  frequency.  That  is,  the  frequency  of 
oscillation  equals  the  impressed  frequency  times  the  square  root 
of  the  ratio  of  condensive  reactance  and  inductive  reactance  of 
the  circuit,  or  is  the  impressed  frequency  divided  by  the  square 
root  of  inductance  voltage  and  capacity  current,  as  fraction  of 
impressed  voltage  and  full-load  current. 
Since 


the  frequency  of  oscillation  is 


that  is,  is  independent  of  the  frequency  of  the  impressed  e.m.f. 
Substituting 

6  =  2  xfi,     xc  =  —^     and      x  =  2  rfL 

&  7T/U 

in  equations  (8),  (10),  and  (11),  we  have 

t 

sin  -    —  y 
VCL 

(12) 


1C       -  —  t  t 

i"  =  V/F#£    2L  cos<90sin 


VCL 

t 


e/'  =-  Ee          cos  ^  cos--—; 


110  TRANSIENT  PHENOMENA 


r    .   ,  .  ,„  t 


1'"  =  e 


VCL 


==•  +  i0  V  7:  sin  — —    [ ; 

C<  T  °  V  /7          A  //nr  r  > 


-~t  ( 
i  =-27zfCEsin(d  -  60)+ £    2L    U0  cos 


(13) 


r 
2L 


e,  =  E  cos  (d  -  00)+  £          I  (e0  -  £  cos  00)  cos  7^=- 

X  ,       «     ) 


(14) 


The  oscillating  terms  of  these  equations  are  independent  of 
the  impressed  frequency.  That  is,  the  oscillating  currents  and 
potential  differences,  caused  by  a  change  of  circuit  conditions 
(as  starting,  change  of  load,  or  opening  circuit),  are  independent 
of  the  impressed  frequency,  and  thus  also  of  the  wave  shape  of 
the  impressed  e.m.f.,  or  its  higher  harmonics  (except  as  regards 
terms  of  secondary  order). 

The  first  component  of  oscillation,  equation  (12),  depends 
not  only  upon  the  line  constants  and  the  impressed  e.m.f.,  but 
principally  upon  the  phase,  or  the  point  of  the  impressed  e.m.f. 
wave,  at  which  the  oscillation  starts;  however,  it  does  not 
depend  upon  the  previous  condition  of  the  circuit.  Therefore 
this  component  of  oscillation  is  the  same  as  the  oscillation 
produced  in  starting  the  transmission  line,  that  is,  connecting 
it,  unexcited,  to  the  generator  terminals. 

There  exists  no  point  of  the  impressed  e.m.f.  wave  where  no 
oscillation  occurs  (while,  when  starting  a  circuit  containing 
resistance  and  inductance  only,  at  the  point  of  the  impressed 
e.m.f.  wave  where  the  final  current  passes  zero  the  stationary 
condition  is  instantly  reached). 

With  capacity  in  circuit,  any  change  of  circuit  conditions 
involves  an  electric  oscillation. 


HIGH  POTENTIAL  SYSTEMS 


111 


The   maximum   intensities  of  the  starting  oscillation  occur 
near  the  value  #0  =  0,  and  are 


and 


Since 


E 


2x&  sin  i/l£ 


xxf 


e/'  =  -  # 


K 


cos  \/_£ 
X 


(15) 


7  = sin  (0  -  00) 


is  the  stationary  value  of  charging  current,  it  follows  that  the 
maximum  intensity  which  the  oscillating  current,  produced  in 

starting  a  transmission  line,  may  reach  is  y  — •  times  the  sta- 
tionary charging  current,  or  the  initial  current  bears  to  the 
stationary  value  the  same  ratio  as  the  frequency  of  oscillation 
to  the  impressed  frequency. 

The  maximum  oscillating  e.m.f.  generated  in  starting  a  trans- 
mission line  is  of  the  same  value  as  the  impressed  e.m.f.  Thus 
the  maximum  value  of  potential  difference  occurring  in  a  trans- 
mission line  at  starting  is  less  than  twice  the  impressed  e.m.f. 
and  no  excessive  voltages  can  be  generated  in  starting  a  circuit. 

The  minimum  values  of  the  starting  oscillation  occur  near 
#0  =  90°,  and  are,  from  equations  (7), 


;„    _  •"    _      2X      COS\/^£ 


and 


-V/-#£. 
"c 


(16) 


that  is,  the  oscillating  current  is  of  the  same  intensity  as  the 
charging  current,  and  the  maximum  rush  of  current  thus  is 
less  than  twice  the  stationary  value.  The  potential  difference 
in  the  circuit  rises  only  little  above  the  impressed  e.m.f. 

The  second  component  of  the  oscillation,  equation  (13),  does 
not  depend  upon  the  point  of  .the  impressed  e.m.f.  wave  at 


112  TRANSIENT  PHENOMENA 

which  the  oscillation  starts,  00,  nor  upon  the  impressed  e.m.f.  as 
a  whole,  E,  but,  besides  upon  the  constants  of  the  circuit,  it 
depends  only  upon  the  instantaneous  values  of  current  and  of 
potential  difference  in  the  circuit  at  the  moment  when  the 
oscillation  starts,  iQ  and  eQ. 

Thus,  if  i0  =  0,  e0  =  0,  or  in  starting  a  transmission  line, 
unexcited,  by  connecting  it  to  the  impressed  e.m.f.,  this  term 
disappears.  It  is  this  component  which  may  cause  excessive 
potential  differences.  Two  cases  shall  more  fully  be  discussed, 
namely : 

(a)  Opening  the  circuit  of  a  transmission  line  under  load,  and 
(6)  rupturing  a  short-circuit  on  the  transmission  line. 

67.  (a)  If  iQ  is  the  instantaneous  value  of  full-load  current, 
e0  the  instantaneous  value  of  difference  of  potential  at  the 
condenser,  n0  is  small  compared  with  e0,  and  \/~xxc  iQ  is  of  the 
same  magnitude  as  e0. 

Writing 


and  substituting  in  equations  (10),  we  have 


**'  cos  0  +  d 


and 


e»>  =  Ve/  +  ia2xxet          sin  U/  ^  0  +  d 


(17) 


e 
that  is,  the  amplitude  of  oscillation  isV/%2  +-  —  for  the  current, 

and  \/eo2  +  i*x xc  for  the  e.m.f.  Thus  the  generated  e.m.f. 
can  be  larger  than  the  impressed  e.m.f.,  but  is,  as  a  rule,  still  of 
the  same  magnitude,  except  when  xc  is  very  large. 

In  the  expressions  of  the  total  current  and  potential  difference 
at  condenser,  in  equations  (11),  (e0  —  E  cos  00)  is  the  difference 
between  the  potential  difference  at  the  condenser  and  the 
impressed  e.m.f.,  at  the  instant  of  starting  of  the  oscillation,  or 
the  voltage  consumed  by  the  line  impedance,  and  this  is  small 


HIGH  POTENTIAL    SYSTEMS 


113 


if  the  current  is  not  excessive.    Thus,  neglecting  the  terms  with 

(e0  —  E  cos  0Q)j  equations  (11)  assume  the  form 


t  -    -      sin  (d  -  60)  + 


v'^'cos  \/-c 

V 


0 


and 


e1=  E  cos  (d  -  60)  +  i0 


-JL0 

ce      ''   siny 


(18) 


that  is,  the  oscillation  of  current  is  of  the  amplitude  of  full-load 
current,  and  the  oscillation  of  condenser  potential  difference  is 
of  the  amplitude  i 


x  xc  is  the  ratio  of  inductance  voltage  to  condenser  current,  in 
fractions  of  full-load  voltage  and  current.     We  have,  therefore, 

L 


Thus  in  circuits  of  very  high  inductance  L  and  relatively  low 
capacity  C,  i0Vx  xc  may  be  much  higher  than  the  impressed 
e.m.f.,  and  a  serious  rise  of  potential  occur  when  opening  the 
circuit  under  load,  while  in  low  inductance  cables  of  high  capacity 
i0Vxxc  is  moderate;  that  is,  the  inductance,  by  tending  to 
maintain  the  current,  generates  an  e.m.f.,  producing  a  rise  in 
potential,  while  capacity  exerts  a  cushioning  effect.  Low 
inductance  and  high  capacity  thus  are  of  advantage  when 
breaking  full-load  current  in  a  circuit. 

68.  (6)  If  a  transmission  line  containing  resistance,  induc- 
tance, and  capacity  is  short-circuited,  and  the  short-circuit 
suddenly  opened  at  time  t  =  0,  we  have,  for  t  <  0, 


and 

where 

and 


ET 

I  =-COS  (d 


-  0Q  -  y), 


tan  y  =  -; 


(19) 


114  TRANSIENT  PHENOMENA 

thus,  at  time  t  =  0, 

_  E 

z 


(20) 


Substituting  these  values  of  e0  and  i0  in  equations  (9)  gives 


and 


E_ 

z 


r_ 

2x 


7      a 

2x 


^-  O 


Xc 


or,  neglecting  terms  of  secondary  magnitude, 


and 


.„,      E  -  £  » 

V       =  —  £  COS 

z 


cos  (8,  +  y)  sin         6; 


(21) 


that  is,  im  is  of  the  magnitude  of  short-circuit  current,  and 
e"'  of  higher  magnitude  than  the  impressed  e.m.f.,  since  z  is 
small  compared  with  \/xxc. 

The  total  values  of  current  and  condenser  potential  difference, 
from  equation  (11),  are 


cos\/- 

X 


xx, 


and 


cos  (0  -  00)  -  Ee 

\rxxc  COS  (00  +  y)    . 


cos 


\  cosV/- 

v    X 


(22) 


HIGH  POTENTIAL   SYSTEMS  115 

or  approximately,  since  all  terms  are  negligible  compared  with 
i'"  and  e/", 


and 


TJI T_Q  i — 

i=-e    2x    cos  (00  +  7-)  cos V- 

2  '    X 

E\fxxc  -£•  fc 
£          cos  (69  +  7-)  sin  y  - 


(23) 


*  •    ju 

These  values  are  a  maximum,  if  the  circuit  is  opened  at  the 
OQ  =  -  r,  that  is,  at  the  maximum  value  of  the  short- 
urrent,  and  are  then 


moment  60  =  -  7-,  that  is,  al 
circuit  current,  and  are  then 


and 


(24) 


The  amplitude  of  oscillation  of  the  condenser  potential  dif 
ference  is 


xx ; 


or,    neglecting   the   line   resistance,    as   rough   approximation, 

x  =  z, 


CE 

x    ' 


that  is,  the  potential  difference  at  the  condenser  is  increased 
above  the  impressed  e.m.f.  in  the  proportion  of  the  square  root 
of  the  ratio  of  condensive  reactance  to  inductive  reactance,  or 
inversely  proportional  to  the  square  root  of  inductance  voltage 
times  capacity  current,  as  fraction  of  the  impressed  voltage  and 
the  full-load  current.  Thus,  in  this  case,  the  rise  of  voltage  is 
excessive. 

The  minimum  intensity  of  the  oscillation  due  to  rupturing 
short-circuit  occurs  if  the  circuit  is  broken  at  the  moment 


116  TRANSIENT  PHENOMENA 

00  =  90°  —  r,  that  is,  at  the  zero  value  of  the  short-circuit  current. 
Then  we  have 


r 

ysiny—  0 

*     £ 


(25) 


that  is,  the  potential  difference  at  the  condenser  is  less  than  twice 
the  impressed  e.m.f.;  therefore  is  moderate.  Hence,  a  short- 
circuit  can  be  opened  safely  only  at  or  near  the  zero  value  of  the 
short-circuit  current. 

The  phenomenon  ceases  to  be  oscillating,  and  becomes  an 
ordinary  logarithmic  discharge,  if  x/r2—  4  xxc  is  real,  or 

r  >  2  Vxx^. 

Some  examples  may  illustrate  the  phenomena  discussed  in  the 
preceding  paragraphs. 

69.  Let,  in  a  transmission  line  carrying  100  amperes  at  full 
load,  under  an  impressed  e.m.f.  of  20,000  volts,  the  resistance 
drop  =  8  per  cent,  the  inductance  voltage  =  15  per  cent  of  the 
impressed  voltage,  and  the  charging  current  =  8  per  cent  of  full- 
load  current.  Assuming  1  per  cent  resistance  drop  in  the 
step-up  transformers,  and  a  reactance  voltage  of  2i  per  cent, 
the  resistance  drop  between  the  constant  potential  generator 
terminals  and  the  middle  of  the  transmission  line  is  then  5  per 
cent,  or  r  =  10  ohms,  and  the  inductance  voltage  is.  10  per 
cent,  or  x  =  20  ohms.  The  charging  current  of  the  line  is  8 
amperes,  thus  the  condensive  reactance  xc  =  2500  ohms. 

Then,  assuming  a  sine  wave  of  impressed  e.m.f.,  we  have 

E  =  20,000  V2  =  28,280  volts; 
i'  =    -  11.3  sin  (6  -  00); 
e{=  28,280  cos  (0  -  00); 

i"  =    -  11.3  £-0'25'[sin  00  cos  11.2  6  -  11.2  cos  0,  sin  11.2  6], 
and     e/'  =  -  28,280  £-°-25'  [cos  00  cos  1 1 .2  0  +  (0.0222  cos  00 

+  0.0283  sin  00)  sin  11. 20] 
=*  -28,280  £-°-25(?  cos  00  cos  11.20. 


HIGH  POTENTIAL   SYSTEMS 


117 


Therefore  the  oscillations   produced  in  starting   the  trans- 
mission line  are 

i  =     -  11.3  [sin  (0  -  00)  +  r0'25  e  (sin  00  cos  11.2  0 

-  11.2  cos  00  sin  11.20)] 
and      e,  =  28,280  { cos  (0  -  00)  -  £-°'25'  [cos  00  cos  11.2  d 

+  (0.0222  cos  00  +  0.0283  sin  00)  sin  1 1 .2  0] } 
^  28,280  [cos  (0  -  00)  -  £-°-25'  cos  00  cos  11.2  0]. 


100        60 

f 

\  s 

"•> 

K=" 
•>•  — 

28,2 
lOo 
20  o 

80  v 
hru 
hm 

ilta 

i  ' 

£-= 

"  25C 

Ooh 

ma 

80  40 
60        30 

V 

\«i 

/ 

\ 

' 

/ 

\ 

(>' 

\ 

/ 

/ 

\ 

/ 

'v 

,- 

40     "SO 

(_ 

\ 

/ 

A^ 

N 

f  o-w-o 

1 

1 

\ 

,' 

/ 

\ 

1 

V 

-  — 

»~--^ 

/ 

\ 

\ 

/ 

, 

\ 

/ 

/ 

\ 

\ 

t 

^r 

•  — 

^y 

i 

-20  10 

1 

p 

\ 

^ 

7 

... 

--\ 

>- 

^ 

I 
1 

1 

\ 
\ 

I 

« 
i 

\, 

/ 

60         30 

1 

/ 

I 
\ 

/ 

on 

1 

/ 

t 

,' 

t 

/ 

inn 

; 

\ 

\ 

/ 

._ 

f 

-120 

\ 

/ 

v 

'' 

20 


70 


40  50 

Degrees 

Fig.  27.     Starting  of  a  transmission  line. 


90 


100 


m 

^^ 


_  —  - 

Zr=> 

•^^ 

• 

^* 

^ 

•*" 

25 

ofl              on 

~-x 

^  —  • 

-z>~* 

^  — 
• 

,~^- 

^* 

s^ 

K  = 

/'  = 

»' 

»hOv 
.hm 

olts 

H 

/; 

\ 

^ 

x^ 

^^^" 

,~v 

Z  = 

x_  — 

•  20< 

jhiu 

lot, 

I 
318 

.10      ^15 

i' 

/ 

U< 

^ 

/ 
/ 

"\ 

•;- 

90 

10      g  10 

a        a 

""x^ 

^ 

-^1 

\ 

/ 

^-—  »  . 

--- 

--. 

\  ^ 

/ 

\ 

^ 

^ 

^X^*^ 

\ 

\ 

\ 

~" 

7- 

-. 

^_ 

\ 

^~ 

/ 

** 

\ 

""  —  • 

""/ 

10           10 

\ 

*  ^^ 

r* 

) 

1 

0 

2 

0 

3 

0 

4 

1 

5 

0 

e 

0 

7 

0 

\ 

0 

I 

0 

1( 

w 

Fig.  28.     Starting  of  a  transmission  line. 

Hence  the  maximum  values  for  00  =  0,  are 

i  --    -11.3  (sin0  -  11.2  £-°-25'  sin  11.20) 
and    e,  =  28, 280  [cos  0-£-°-25'  (cos  11.2  0  +  0.0222  sin  11.20)] 

^  28,280  (cos  0  -  £-°-25'  cos  11.2  0), 
and  the  minimum  values,  for  00  =  90°,  are 

i  -=  11.3  (cos  0  -  <r°-25'cos"11.20) 
and    e,  =  28,280  (sin  0-0.0283  i~°'250  sin  11.2  0) 
s  28,280  sin  0 


118 


TRANSIENT  PHENOMENA 


These  values  are  plotted  in  Figs.  27  and  28,  with  the  current,  i, 
in  dotted  and  the  potential  difference,  ely  in  drawn  line.  The 
stationary  values  are  plotted  also,  in  thin  lines,  i  and  e',  respec- 
tively. 

(a)  Opening  the  circuit  under  full  load,  we  have 

i  =  -  11.3  sin  (6  -  00)  +  iQe-°-**9  cos  11.2  6 
and  el  =  28,280  cos  (6  -  00)  +  224  v^^'sin  11.2/9. 


-100 


a  10— £-20 
«  o-J-o 

-10 — W-20 
-20 40 


-100 
-120 


i0=fl414  amp. 


10 


50 


80 


100 


Fig.  29.     Opening  a  loaded  transmission  line. 

These  values  are  maximum  for  00  =  0  and  non-inductive 
circuit,  or  i0  =  141.4,  and  are 

i  =  -11.3  sin  6  +  141.4  r0'26' cos  11.2  6 
and  el  =  28,280cos0  +  31,600£-°-25*  sin  11.20. 

These  values  are  plotted,  in  Fig.  29,  in  the  same  manner  as 
Figs.  27  and  28. 

(6)  Rupturing  the  line  under  short-circuit,  we  have 


and 

and  therefore 


z  =  22.4 

i0  =  1265  cos  (00  +  r); 

11.3  sin  (6  -  00)  +  1265  r°'259 [cos  (00  +  r) 
cos  11.2  0  +  0.1  cos  dn  sin  11.2  d] 


HIGH  POTENTIAL  SYSTEMS  119 

and    e1  =  28,280  Scos  (0  -  00)  -  s-°'250[cos  00  cos  11.2  0 

-  10  cos  (00  -f  r)  sin  11.2  0]j. 

These  values  are  a  maximum  for  00  =  -  7-  =  -  63°,  thus 

i  =  -  11.3  sin  (0  +  63°)  +  1265  e"°'25d  (cos  11.2  0 
+  0.044  sin  11.2  0) 

and     el  =  28,280  cos  (0  +  63°)  -  282,800  rQ'25d  (0.044  cos  11.2  0 

-  sin  11.20); 

that  is,  the  potential  difference  rises  about  tenfold,  to  282,800 
volts.    These  values  are  plotted  in  Fig.  30. 


-1200 


100 


Fig.  30.     Opening  a  short-circuited  transmission  line. 

70.  On  an  experimental  10,000-volt,  40-cycle  line,  when  a 
destructive  e.m.f.  was  produced  by  a  short-circuiting  arc,  the 
author  observed  a  drop  in  generator  e.m.f.  to  about  5000  volts, 
due  to  the  limited  machine  capacity.  The  resistance  of  the 
system  was  very  low,  about  r  =  1  ohm,  while  the  inductive 
reactance  may  be  estimated  as  x  .=  10  ohms,  and  the  condensive 
reactance  as  xc  =  20,000  ohms.  Therefore  tan  f  =  10,  or 
approximately,  7-  =  90°. 

Herefrom  it  follows  that 


and 


i  =  707  r0'05* cos  44.70 
el  =  316,000  r^'sin  44.70; 


120  TRANSIENT  PHENOMENA 

that  is,  the  oscillation  has  a  frequency  of  about  1800  cycles  per 
second  and  a  maximum  e.m.f.  of  nearly  one-third  million  volts, 
which  fully  accounts  for  its  disruptive  effects. 
71.  As  conclusion,  it  follows  herefrom: 

1.  A    most    important    source  of    destructive  high  voltage 
phenomena  in  high  potential  circuits  containing  inductance  and 
capacity  are  the  electric  oscillations  produced  by  a  change  of 
circuit  conditions,  as  starting,  opening  circuit,  etc. 

2.  These  phenomena  are  essentially  independent  of  the  fre- 
quency and  the  wave  shape  of  the  impressed  e.m.f.,  but  de- 
pend upon  the  conditions  under  which  the  circuit  is  changed, 
as  the  manner  of  change  and  the  point  of  the  impressed  e.m.f. 
and  current  wave  at  which  the  change  occurs. 

3.  The  electric  oscillations  occurring  in  connecting  a  trans- 
mission line  to  the  generator  are  not  of  dangerous  potential,  but 
the  oscillations  produced  by  opening  the  transmission  circuit 
under  load  may  reach  destructive  voltages,  and  the  oscillations 
caused  by  interrupting  a  short-circuit  are  liable  to  reach  voltages 
far  beyond  the  strength  of  any  insulation.    Thus  special  pre- 
cautions should  be  taken  in  opening  a  high  potential  circuit 
under  load.     But  the  most  dangerous  phenomenon  is  a  low 
resistance  short-circuit  in  open  space. 

4.  The  voltages  produced  by  the  oscillations  in  open-circuiting 
a  transmission  line  under  load  or  under  short-circuit  are  mod- 
erate if  the  opening  of  the  circuit  occurs  at  a  certain  point  of 
the  e.m.f.  wave.    This  point  approximately  coincides  with  the 
moment  of  zero  current. 


CHAPTER  IX. 


DIVIDED    CIRCUIT. 

72.  A  circuit  consisting  of  two  branches  or  multiple  circuits 
1  and  2  may  be  supplied,  over  a  line  or  circuit  3,  with  an  impressed 
e.m.f.,  e0. 

Let,  in  such  a  circuit,  shown  diagrammatically  in  Fig  31, 
rv  Lv  Cl  and  r2,  L2,  Cz  —  resistance,  inductance,  and  capacity, 
respectively,  of  the  two  branch  circuits  1  and  2;  r0,  L0,  C0  = 

Co 


Fig.  31.     Divided  circuit. 

resistance,  inductance,  and  capacity  of  the  undivided  part  of  the 
circuit,  3.  Furthermore  let  e  =  potential  difference  at  terminals 
of  branch  circuits  1  and  2,  it  and  i2  respectively  =  currents  in 
branch  circuits  1  and  2,  and  i3  =  current  in  undivided  part  of 
circuit,  3. 

Then  ia  =  il  +  i2 

and  e.m.f.  at  the  terminals  of  circuit  1  is 


of  circuit  2  is 


e  = 


di 


121 


(2) 


(3) 


122  TRANSIENT   PHENOMENA 

and  of  circuit  3  is 

(4) 

Instead  of  the  inductances,  L,  and  capacities,  C,  it  is  usually 
preferable,   even  in  direct-current    circuits,   to    introduce  the 

reactances,  x  =  2  nfL  =  inductive  reactance,  xc  =  =  con- 

2  7T/G 

densive  reactance,  referred  to  a  standard  frequency,  such  as 
/  =  60  cycles  per  second.     Instead  of  the  time  t,  then,  an  angle 

0  =  2  nft  (5) 

is  introduced,  and  then  we  have 

di        x    di  dd         di 

^  *  (6) 


i/iift  -  2  Kfo  f  Id0  -  Xc/i  dd, 


t 
since 


Hereby  resistance,  inductance,  and  capacity  are  expressed  in 
the  same  units,  ohms. 

Time  is  expressed  by  an  angle  6  so  that  360  degrees  correspond 
to  sV  of  a  second,  and  the  time  effects  thus  are  directly  com- 
parable with  the  phenomena  on  a  60-cycle  circuit. 

A  better  conception  of  the  size  or  magnitude  of  inductance 
and  capacity  is  secured.  Since  inductance  and  capacity  are 
mostly  observed  and  of  importance  in  alternating-current  cir- 
cuits, a  reactor  having  an  inductive  reactance  of  x  ohms  and 
i  amperes  conveys  to  the  engineer  a  more  definite  meaning  as 
regards  size:  it  has  a  volt-ampere  capacity  of  tfx,  that  is,  the 
approximate  size  of  a  transformer  of  half  this  capacity,  or  of  a 

^2x 

—  -watt  transformer.    A  reactor  having  an  inductance  of  L 

henrys  and  i  amperes,  however,  conveys  very  little  meaning  to 


DIVIDED  CIRCUIT  123 

the  engineer  who  is  mainly  familiar  with  the  effect  of  inductance 
in  alternating-current  circuits. 

Substituting  therefore  (5)  and  (6)  in  equations  (2),  (3),  (4), 
gives  the  e.m.f.  in  circuit  1  as 

dL 
e  =  rli1  +  xl  -r1  +  a 

in  circuit  2  as 

dL        '    C 
*  =  r**  +  **-fi  +  *ctJi,M',  (8) 

in  circuit  3  as 

e  =  e  a.  r  {   4.  x  -h.  _j_  x    I  {  ^.  /Q\ 

0  03      '          0    J/j      '         CQ    I      3          J  v*^/ 

tZC7  «/ 

hence,  the  potential  differences  at  the  condenser  terminals  are 

/di< 
i,dd  =  e-r1i1-  xt— S  (10) 

e2=  <J  *i  dd  =  e-  r2i2  -*,-^>  (11) 

and  e3  =  xco  I  i3dd  =  e0-  e  -  r0i3  -  x0  -^  •          (12) 

Differentiating  equations  (7),  (8),  and  (9),  to  eliminate  the 
integral,  gives  as  differential  equations  of  the  divided  circuit: 

d?ii         dil          .       de 
d  in         din  de 


cPt'  .       di~          .       deQ      de 

and  +  r*--- 


Subtracting  (14)  from  (13)  gives 

d\\          di,  .\     /     d?i2          di2 


124  TRANSIENT  PHENOMENA 

Multiplying  (15)  by  2,  and  adding  thereto  (13)  and  (14),  gives, 
by  substituting  (1),  i3  =  it  +  t'2, 

(]    i  fi'l 

(2  x0  +  x,)  -^  +  (2  r,  +  r^+  (2  xco  +  xji,  + 

(2  *0  +  x,)  J|  +  (2  r0  +  r2)  ^  +  (2  *Co  +  xc>'2  =  2  ^ .   (17) 

These  two  differential  equations  (16)  and  (17)  are  integrated 
by  the  functions 

and  -  (18) 

i2  =  i2'  +  A2e~ae, 

where  if  and  i2    are  the  permanent  values  of   current,  and 
i"  =  A^~a9  and  i2"  =  A2e~ae  are  the  transient  current  terms. 
Substituting  (18)  in  (16)  and  (17)  gives 

/-/2/j    /  sl/\     * 

\Jj      t/o  (JLlci 


+  A^-a9  (a\  -  ar1  +  xc)  -  A2e~a0  (a?x2  -  ar2  +  xc)  =  0      (19) 

and 

(Pi  '  di'  - 


(2  r0  +  r,)         +  (2'^  +  xc)i 


-  a  (2  r0  +  r,)  +  (2  zco  +  xCl)}  +  A^-^  {a2  (2  x0  +  xa) 
-  a  (2r0  +  r2)  +  (2  xco  +  xc)}  -2^-  (20) 

73.  For  0  =  oo,  the  exponential  terms  eliminate,  and  there 
remain  the  differential  equations  of  the  permanent  terms 
i/  and  t/,  thus 


0    (21) 

and 

cPt  '  di  r 

(2^o  +  x,)  -^-  +  (2r0  +  r,)  ^-  +  (2  xco  +  ajj  i/  +  (2  a0  +  x2) 

^  +(2r   +r)-^+(2z+zH'-2^.  C22) 

2        ^  ro  -t  r2;  k    ^co+  *C2;  ^2   -  * 


DIVIDED   CIRCUIT  125 

The  solution  of  these  equations  (21)  and  (22)  is  the  usual 
equation  of  electrical  engineering,  giving  t/  and  i2  as  sine  waves 
if  the  e.m.f.,  e0,  is  a  sine  wave;  giving  ^'1/  and  i2  as  constant 
quantities  if  e0  is  constant  and  xco  and  either  xCi  or  xct  or  both 
vanish,  and  giving  i{  and  i2  =  0  if  either  xco  or  both  xCl  and 
xCt  differ  from  zero. 

Subtracting  (21)  and  (22)  from  (19)  and  (20)  leaves  as  dif- 
ferential equations  of  the  transient  terms  i"  and  i"t 

e~ae  {A,  (a\  -  or,  +  xci)  '-  A2  (a?x2  -  ar2  +  xj}  -=  0     (23) 
and 

e-«*  {A,  [a2  (2  x0  +  «,)•-?  (2  ro  +  'i)  +  (2  zco  +  xci)]  +  A2 

[a2  (2  z0  +  x2)  -  a  (2r0  +  r2)  +  (2  xco  +xC2)]}  =  0.  (24) 

Introducing  a  new  constant  5,  these  equations  give,  from  (23), 
A1  =  B  (a?x2  -  ar2  +  xc) 


and 


(25) 


then  substituting  (25)  in  (24)  gives 

(a?x2  -  ar2  +  xc)  [a2  (2  x0  +  xj  -  a  (2  r0  +  r,)  +  (2  ^Co  +  xj] 
+  (a2^,  -  ort  +  xci)[a2(2  x0  +  x2)  -  a  (2  r0  +  r2)  +  (2  xco 
+  *c2)]  =  0,  (26) 

while  B  remains  indeterminate  as  integration  constant. 

Quartic  equation  (26)  gives  four  values  of  a,  which  may  be  all 
real,  or  two  real  and  two  conjugate  imaginary,  or  two  pairs  of 
conjugate  imaginary  roots. 

Rearranged,  equation  (26)  gives 


a4  (XQX,  +  x0x2  +  x,x2)  -  a3  {r0  (xl  +  x2)  +  r1  (x,  -f  x2) 
+  r2  (x0  +  x^)  }  +  a2  {  (r/!  +  r/a  +  r/,)  +  xCo  (x1  +  x2) 
+  xci  (x0+  x2)+  xcz  (XQ+  x,)}-  a  {xeo(rl+  r2)+  xCi  (r0-f  ra) 
•  +  ^c,  (ro  +  ri)  }  +  (  VCl  +  XA*  +  ^c^c.)  -  0.  (27) 

Let  o17  a2,  a3,  a4  be  the  four  roots  of  this  quartic  equation  (27)  ; 


126 

then 

and 

i2  =  i 


B 


TRANSIENT  PHENOMENA 


B 


-  a/2  +  xc) 


xc 


B4  (a*x2-  a4r2+ 
^  +  B    a2x  -  a 


(28) 


(29) 


where  the  integration  constants  Bv  B2,  B3  and  B4  are  deter- 
mined by  the  terminal  conditions:  the  currents  and  condenser 
potentials  at  zero  time,  6  =  0. 

The  quartic  equation  (27)  usually  has  to  be  solved  by  approxi- 
mation. 

74.  Special  Cases:  Continuous-current  divided  circuit,  with 
resistance  and  inductance  but  no  capacity,  e0  =  constant. 


Fig.  32.     Divided  continuous-current  circuit  without  capacity. 

In  such  a  circuit,  shown  diagrammatically  in  Fig.  32,  equations 
(7),  (8),  and  (9)  are  greatly  simplified  by  the  absence  of  the 
integral,  and  we  have 

e~ '&  +  *&>  (30) 


and  eQ  =  e  -\ 

(30)  and  (31)  combined  give 


di. 
1- 


.  3 

'°d0 


- 


(31) 
(32) 

(33) 


DIVIDED  CIRCUIT  127 

Substituting  (1),  i3  =  i1  +  iv  in  (32),  multiplying  it  by  2  and 
adding  thereto  (30)  and  (31),  gives 

2  e0=  (2  r0+  rt)  t1  +  (2  r0+  r2)  *,+  (2  z0  +  xt)  ^ 


Equations  (33)  and  (34)  are  integrated  by 
and  F  (35) 

Substituting  (35)  in  (33)  and  (34)  gives 
(VV  -  r2V)  +  e-^Ufa-  ax,)  -  A2(r2  -  ax2)l  =  0 

and 

2  e0=  (2  r0  +  rt)  i/  +  (2  r0  +  ra)  i/  +  e~ai{Al  [(2  r0  +  rf) 

-  a  (2  z0  +  x,)]  +  A2  [(2  r0  +  r2)-  a  (2  x0  +  *,)]}. , 

These  equations  resolve  into   the  equations  of  permanent 
state,  thus 

and  (2  r0  +  r,)  i/  +  (2  r0  +  ra)  t/  =  2  e0.  J 

Hence,  t/  =  60-^ 

•(36) 
and  t'2'  =  e0-j? 

where  r2  =  r0rt  +  r/2  +  rtr2,  (37) 

and  the  transient  equations  having  the  coefficients 

A     (r          nv\-     A     (r    --  nr  "I    —  0 

Ai  yt      *f*v      A2  vr2      ax2/ 
and 

At  [(2  r0  +  rt)  -  a  (2  x0  +  xj]  +  A2  [(2  r0  +  r2) 
-  a  (2  z0  +  xa)]  =  0. 


128  TRANSIENT  PHENOMENA 

Herefrom  it  follows  that 


and 
and 


A2  =  B  (r1  -  axj, 


a2 


x,)] 


-  a[r0  (xl  +  x2) 


=  0, 


x2) 


5  -  indefinite. 
Substituting  the  abbreviations, 


(38) 


(39) 
(40) 


and 


\  +  V2  +  rjr2  =  r2, 
r0  (^i  +  sa)  +  ri  (^o  +  ^2)  +  r2  (*0  +  ^i)  =  ZG  (ri  +  r2) 


^2  (r 


gives  (39) 
hence  two  roots, 

and 
where 


=  0; 


2X2 


°2  =  "r^ 


-  4  rV. 


(4D 


(42) 


(43) 


(44) 


The  two  roots  of  equation  (42),  ax  and  a2,  are  always  real,  since 
in  (f 

s4  >  4  rV, 


as  seen  by  substituting  (41)  therein. 
The  final  integral  equations  thus  are 


and 


2x2 


*+<f 


-2 


(45) 


DIVIDED   CIRCUIT 


129 


B^  and  B2  are  determined  by  the  terminal  conditions,  as  the 
currents  ^  and  t'2  at  the  start,  6  =  0. 
Let,  at  zero  time,  or  0  =  0, 


and 


then,  substituting  in  (45),  we  have 


(46) 


4- 

l  1 


and 


(47) 


and  herefrom  calculate  Bl  and  £2. 

75.  For  instance,  in  a  continuous-current  circuit,  let  the 
impressed  e.m.f.,  e0  ==  120  volts;  the  resistance  of  the  undivided 
part  of  the  circuit,  r0  —  20  ohms;  the  reactance,  x0  =  20  ohms; 
the  resistance  of  one  of  the  branches,  rl  =  20  ohms;  the  reactance, 
xl  =  40  ohms,  and  the  resistance  of  the  other  branch,  r2  = 
5  ohms,  the  reactance,  x2  =  200  ohms. 

Thus  one  of  the  branches  is  of  low  resistance  and  high  react- 
ance, the  other  of  high  resistance  and  moderate  reactance. 

The  permanent  values  of  the  currents,  (r2  =  600),  are 


and 


t/  =  1  amp. 


i 


i2   =  4  amp. 

(a)  Assuming  now  the  resistance  r0  suddenly  decreased  from 
ro  =  20  ohms  to  r0  =  15  ohms,  we  have  the  permanent  values 
of  current  as 

and 


t/  =  1.265  amp. 
if  =  5.06  amp. 


The  previous  values  of  currents,  and  thus  the  values  of  currents 
at  the  moment  of  start,  0  =  0,  are 

i*  =  1  amp.  1 

and  r 

i2°  =  4  amp.  J 


130  TRANSIENT  PHENOMENA 

therefrom  follow  the  equations  of  currents,  by  substitution  in 
the  preceding, 


i,  =  1.265  +  0.455  £-<>'<™3e  -  0.720  e~0'^6  1 
and 

i2  -  5.06  -  1.038  r0'0633*  _  0.022  e"  °'586'.    J 

(6)  Assuming  now  the  resistance  r0  suddenly  raised  again 
from  r0  =  15  ohms  to  r0  =  20  ohms,  leaving  everything  else 
the  same,  we  have 

ij0  =  1.265  amp.        ~] 
and 

t'2°  =  5.06  amp.;        J 

and  then 


i,  =  1  -  0.528  e-0'06970  +  0.793  r0'674'  1 

and 

ia  =  4  +  1.018  s-0-069™  +  0.042  £-°'6740  .          J 

(c)  Assuming   now  the   resistance    r0  suddenly  raised    from 
r0  =  20  ohms  to  r0  =  25  ohms,  gives 


i,  =  0.828  -  0.374  e-Q'9™e  +  0.546  £-°'7640      1 
and 

i2  =  3.312  +  0.649  rQ'm39  +  0.039  r0-764'.     j 

(^)  Assuming  now  the  resistance  r0  lowered  again  from  r0  = 
25  ohms  to  r0  =  20  ohms,  gives 

i,  =  1  +  0.342  r0-069™  -  0.514  r0'6740  ^ 

and  L 

ia  =  4  -  0.660  g-0-06970  _  0.028  £-°-674'.  J 

76.  In  Fig.  33  are  shown  the  variations  of  currents  il  and  iv 
resultant  from  a  sudden  variation  of  the  resistance  r0  from  20 
to  15,  back  to  20,  to  25,  and  back  again  to  20  ohms.  As  seen, 
the  readjustment  of  current  iv  that  is,  the  current  in  the  induc- 
tive branch  of  the  circuit,  to  its  permanent  condition,  is  very 
slow  and  gradual.  Current  iv  however,  not  only  changes  very 
rapidly  with  a  change  of  r0,  but  overreaches  greatly;  that  is,  a 
decrease  of  r0  causes  il  to  increase  rapidly  to  a  temporary  value 
far  in  excess  of  the  permanent  increase,  and  then  gradually  it 


DIVIDED  CIRCUIT 


131 


falls  back  to  its  normal,  and  inversely  with  an  increase  of  r0. 
Hence,  any  change  of  the  main  current  is  greatly  exaggerated 
in  the  temporary  component  of  current  i\;  a  permanent  change 
of  about  20  per  cent  in  the  total  current  results  in  a  practically 
instantaneous  change  of  the  branch  current  iv  by  about  50  per 
cent  in  the  present  instance. 

Thus,  where  any  effect  should  be  produced  by  a  change  of 
current,  or  of  voltage,  as  a  control  of  the  circuit  effected  thereby, 
the  action  is  made  far  more  sensitive  and  quicker  by  shunting 
the  operating  circuit  iv  of  as  low  inductance  as  possible,  across 


t=-0       20       40 


Fig.  33.     Current  in  divided  continuous-current  circuit  resulting  from  sudden 
variations  in  resistance. 

a  high  inductance  of  as  low  resistance  as  possible.  The  sudden 
and  temporary  excess  of  the  change  of  current  il  takes  care  of 
the  increased  friction  of  rest  in  setting  the  operating  mechanism 
in  motion,  and  gives  a  quicker  reaction  than  a  mechanism 
operated  directly  by  the  main  current. 

This  arrangement  has  been  proposed  for  the  operation  of  arc 
lamps  of  high  arc  voltage  from  constant  potential  circuits. 
The  operating  magnet,  being  in  the  circuit  iv  more  or  less 
anticipates  the  change  of  arc  resistance  by  temporarily  over- 
reaching. 

77.  The  temporary  increase  of  the  voltage,  e,  across  the 
branch  circuit,  iv  corresponding  to  the  temporary  excess  current 
of  this  circuit,  may,  however,  result  in  harmful  effects,  as  de- 
struction of  measuring  instruments  by  the  temporary  excess 
voltage. 


132  TRANSIENT  PHEXOMEXA 

Let,  for  instance,  in  a  circuit  of  impressed  continuous  e.m.f., 
^  =  600  volts,  as  an  electric  railway  circuit,  the  resistance  of 
the  circuit  equal  25  ohms,  the  inductive  reactance  44  ohms. 
This  gives  a  permanent  current  of  i7  =  24  amperes. 

Let  now  a  small  part  of  the  circuit,  of  resistance  ra  =  1  ohm, 
but  including  most  of  the  reactance  x,  =  40  ohms  —  as  a  motor 
series  field  winding  —  be  shunted  by  a  voltmeter,  and  rl  =  1000 
ohms  =  resistance,  xx  =  40  ohms  =  reactance  of  the  volt- 
meter circuit. 

In  permanent  condition  the  voltmeter  reads  ,V  X  600  =  24 
volts,  but  any  change  of  circuit  condition,  as  a  sudden  decrease 
or  increase  of  supply  voltage  ev  results  in  the  appearance  of  a 
temporary  term  which  may  greatly  increase  the  voltage  impressed 
upon  the  voltmeter. 

In  this  divided  circuit,  the  constants  are:  undivided  part  of 
the  circuit,  rt  =  24  ohms;  xt  =  4  ohms;  first  branch,  voltmeter 
(practically  non-inductive),  rt  =  1000  ohms,  x4  =  40  ohms; 
second  branch,  motor  field,  highly  inductive,  r,  =  1  ohm,  x,  = 
40  ohms. 

(a)  Assuming  now  the  impressed  e.m.f.,  ev  suddenly  dropped 
from  e%  =  600  volts  to  e%  =  540  volts,  that  is,  by  10  per  cent, 
gives  the  equations 


•ad 


i,  =  0.0216  -  0.0806  £-••""  +  0.0830 £-*-lf| 
t,  =  21.6  +  2.407  £-••«*•  -  0.007  £-*Jf . 


(5)  Assuming  now  the  voltage,  ev  suddenly  raised  again  from 
et  =  540  volts  to  c.  =  600  volts,  gives  the  equations 

i,  =  0.024  +  0.0806  £-••«•-  0.0830  £- 
and 

i,  =  24  -  2.407  e-*402*  +  0.007 


.0830  £-s-lfl 

r*'1'  . 


Tbe  voltage,  e,  across  the  voltmeter,  or  on  circuit  1,  is 
e  =  r^  +  *i^r  =  1000 1/  =F  77.9 e~9jm§  ±  6.2 r**9 , 

where 


DIVIDED  CIRCUIT 


133 


Hence,  in  case  (a),  drop  of  impressed  voltage,  ev  by  10  per  cent, 

e  =  21.6  -  77.9  £-••""+  6.2 
and  in  (6),  rise  of  impressed  voltage, 

e  =  24.0  +  77.9  «-•"""-  6.2 

This  voltage,  e,  in  the  two  cases,  is  plotted  in  Fig.  34.  As 
seen,  during  the  transition  of  the  voltmeter  reading  from  21.6 
to  24.0  volts,  the  voltage  momentarily  rises  to  95.7  volts,  or 


Fig. 


jve  apparatus  in  series  with  cdrcuit  of  high 
mniirtann* 


four  times  its  permanent  value,  and  during  the  decrease  of 
permanent  voltage  from  24.0  to  21.6  volts  the  voltmeter  momen- 
tarily reverses,  going  to  50.1  volts  in  reverse  direction. 

In  a  high  voltage  direct-current  circuit,  a  voltmeter  shunted 
across  a  low  resistance,  if  this  resistance  is  highly  inductive,  is  in 
danger  of  destruction  by  any  sudden  change  of  voltage  or  current 
in  the  circuit,  even  if  the  permanent  value  of  the  voltage  is  well 
within  the  safe  range  of  the  voltmeter. 


CAPACITY  SHUNTING  A  PART  OF  A  OONTINUOU^CURRENT 

CIRCUIT. 

78.  A  circuit  of  resistance  rt  and  inductive  reactance  xl  is 
shunted  by  the  condensive  reactance  xc,  and  supplied  over  the 
resistance  r0  and  the  inductive  reactance  x.  by  a  continuous 
impressed  e.m.f.,  ^,  as  shown  diagrammatically  in  Fig.  35. 


134  TRANSIENT  PHENOMENA 

In  the  undivided  circuit, 


di.      di 


In  the  inductive  branch, 


In  the  condenser  branch, 
e  =  xc  I  i2  dd. 


(48) 


(49) 


(50) 


L0, 


Fig.  35.     Suppression  of  pulsations  in  direct-current  circuits  by  series  induc- 
tance and  shunted  capacity. 

Eliminating  e  gives,  from  (48)  and  (49), 

fl'l  {11 


and  from  (49)  and  (50), 


(52) 


Differentiating  (52),  to  eliminate  the  integral, 


(53) 


DIVIDED  CIRCUIT  135 


Substituting  (53)  in  (51),  and  rearranging, 


1  (  di 

eo  =   fro  +  ri)  h  +  -      frori  +  Vo  +  3e*i)  -Z 


i  +  r^o)  ^  +  XO*I-J£JT  \  > 


a  differential  equation  of  third  order. 
This  resolves  into  the  permanent  term 

<>0    =    0*0    +   O  \', 

hence,  i,'  =  —  ^—  ,  (55) 

TQ  H-  rt 

and  a  transient  term 

t,"  -  4«-»;  (56) 

that  is, 

^  -  V  +  A£~^  =  —  ^—  +  Ae-ae.  (57) 

r0  +  rt 

Equation  (57)  substituted  in  (54)  gives  as  equation  of  a, 

*c  fro  +  ri)  ~  a  frori  +  «A  +  %)  +  °2  fro^i  +  ri^o)  -  a'^i  =  °> 
or 

^L+i1)  =  0    (58) 


while  A.  remains  indefinite  as  integration  constant. 

Equation  (58)  has  three  roots,  av  a2,  and  a3,  which  either  are 
all  three  real,  when  the  phenomenon  is  logarithmic,  or,  one 
real  and  two  imaginary,  when  the  phenomenon  is  oscillating. 

The  integral  equation  for  the  current  in  branch  1  is 


*e  •         (59) 
the  current  in  branch  2  is  by  (53) 


(60) 


136  TRANSIENT  PHENOMENA 

and  the  potential  difference  at  the  condenser  is 

/dfL 
i2dd=rlil  +  z,^1 


"'.  (61) 

In  the  case  of  an  oscillatory  change,  equations  (59),  (60),  and 
(61)  appear  in  complex  imaginary  form,  and  therefore  have  to 
be  reduced  to  trigonometric  functions. 

The  three  integration  constants,  Av  A2,  and  A3,  are  deter- 
mined by  the  three  terminal  conditions,  at  6  =  0,  il  =  if, 
iz  =  if,  e  =  e°. 

79.  As  numerical  example  may  be  considered  a  circuit  having 
the  constants,  e0  =  110  volts;  r0  =  1  ohm;  x0  =  10  ohms; 
rl  =  10  ohms;  x^  =  100  ohms,  and  xc  =  10  ohms. 

In  other  words,  a  continuous  e.m.f.  of  110  volts  supplies, 
over  a  line  of  r0  =  1  ohm  resistance,  a  circuit  of  rl  =  10  ohms 
resistance.  An  inductive  reactance  x0  =  10  ohms  is  inserted 
into  the  line,  and  an  inductive  reactance  xl  =  100  ohms  in  the 
load  circuit,  and  the  latter  shunted  by  a  condensive  reactance  of 
xc  =  10  ohms. 

Then,  substituting  in  equation  (58), 

a3  -  0.2  a2  +  1.11  a  -  0.11  =  0. 

This  cubic  equation  gives  by  approximation  one  root,  at  =  0.1, 
and,  divided  by  (a  —  0.1),  leaves  the  quadratic  equation 

a2  -  0.1  a  +  1.1  =  0, 

which  gives  the  complex  imaginary  roots  a2  =  0.05  —  1.047  / 
and  a3  =  0.05  +  1.047  j;  then  from  the  equation  of  current, 
by  substituting  trigonometric  functions  for  the  exponential 
functions  with  imaginary  exponent,  we  get  the  equation  for  the 
load  current  as 

i,  =  if  +  A^-*'10  +  £-°-05'  (Bl  cos  1.047  6  +  B2  sin  1.047  0), 
the  condenser  potential  as 

e=W  if  +  e-0'050  {(5^  +  104.7  B2)  cos  1.047  d  -  (104.7  Bl 
-  5  £2)  sin  1.047/9}, 


DIVIDED  CIRCUIT  137 

and  the  condenser  current  as 

ia  =  10.9  r0-05'  {B,  cos  1.047  0  +  B2  sin  1.047  6}. 

At  e0  =  110  volts  impressed,  the  permanent  current  is  if  =  10 
amp.,  the  permanent  condenser  potential  is  ef  =  100  volts,  and 
the  permanent  condenser  current  is  if  =  0. 

Assuming  now  the  voltage,  e0,  suddenly  dropped  by  10  per 
cent,  from  e0  =  110  volts  to  e0  =  99  volts,  gives  the  permanent 
current  as  if  =  9  amp.  At  the  moment  of  drop  of  voltage, 
0  =  0,  we  have,  however,  il  =  if  =  10  amp.;  e  =  e'  =  100 
volts,  and  t'2  =  0;  hence,  substituting  these  numerical  values 
into  the  above  equations  of  iv  e,  iv  gives  the  three  integration 
constants : 

A,  =  1;  Bi  =  0,  and  B2  =  0.0955; 
therefore  the  load  current  is 

i,  =  9  +  r°'ie  +  0.0955  r0'05'  sin  1.047  0, 
the  condenser  current  is 

i2  =  1.05  r0'05'  sin  1.0470, 
and  the  condenser,  or  load,  voltage  is 

e  =  90  +  e-0'06*  (10  cos  1.047  0  +  0.48  sin  1.047  0). 
Without  the  condenser,  the  equation  of  current  would  be 
i=9  +  r°'ie. 

In  this  combination  of  circuits  with  shunted  condensive 
reactance  xc,  at  the  moment  of  the  voltage  drop,  or  0  =  0,  the 
rate  of  change  of  the  load  current  is,  approximately, 

^  =  [_  O.lr0'1'  +  0.0955  X  1.047r°-05e  cos  1.047  0]0  =  0, 
do 

while  without  the  condenser  it  would  be 

|  =  [_  o.l  r  «•"].  =  -0.1. 

80.  By  shunting  the  circuit  with  capacity,  the  current  in  the 
circuit  does  not  instantly  begin  to  change  with  a  change  or 
fluctuation  of  impressed  e.m.f. 


138 


TRANSIENT  PHENOMENA 


In  Fig.  36  is  plotted,  with  6  as  abscissas,  the  change  of  the 
current,  ilt  in  per  cent,  resulting  from  an  instantaneous  change 
of  impressed  e.m.f.,  eQ,  of  10  per  cent,  with  condenser  in  shunt 
to  the  load  circuit,  and  without  condenser. 

As  seen,  at  0  =  172°,  both  currents,  it  with  the  condenser 
and  i  without  condenser,  have  dropped  by  the  same  amount, 


2.6 
2.4 
2.2 
2.0 
1.8 
1.6 
1.4 

1.0 

o.s 

0.6 
0.4 
0.2 
0 

j 

upply: 

co- 

=  110  volts 

/I 

1  0  = 

=     1  ohm 
=  10  ohms 

X 

/ 

/ 

Lc 

id  i 

nd 

series  i'ndu'ctan 
l\  ==  10  ohms 

ce 

/ 

/ 

Xi  i  100  oh 

tns 

/ 

/ 

SI 

nut 

(_•<!  c 

apa 

e-ily 
--   1C 

oh 

IIS 

/ 

/ 

^ 

X 

/ 

V 

^ 

/ 

/ 

/ 

'</ 

A 

/ 

/ 

'^£> 

j 

// 

/ 

*i\ 

• 

/ 

^ 

' 

/ 

/ 

/ 

—  —• 

*^ 

^ 

0.4 


1.2 


1.6 


2.0 


2.4 


2.8 


Fig.  36.     Suppression  of  pulsations  in  direct-current  circuits  by  series  induc- 
tance and  shunted  capacity.     Effect  of  10  per  cent  drop  of  voltage. 

2.6  per  cent.  But  at  0  =  57.3°,  ^  has  dropped  only  J  per  cent, 
and  i  nearly  1  per  cent,  and  at  0  =  24°,  il  has  not  yet  dropped 
at  all,  while  i  has  dropped  by  0.38  per  cent. 

That  is,  without  condenser,  all  pulsations  of  the  impressed 
e.m.f.,  eQ,  appear  in  the  load  circuit  as  pulsations  of  the  current, 
i,  of  a  magnitude  reduced  the  more  the  shorter  the  duration  of 
the  pulsation.  After  0  =  60°,  or  t  =  0.00275  seconds,  the 
pulsation  of  the  current  has  reached  10  per  cent  of  the  pulsation 
of  impressed  e.m.f. 

With  a  condenser  in  shunt  to  the  load  circuit,  the  pulsation 
of  current  in  the  load  circuit  is  still  zero  after  0  =  24°,  or  after 
0.001  seconds,  and  reaches  1.25  per  cent  of  the  pulsation  of 
impressed  e.m.f.,  e0,  after  6  =  60°,  or  t  =  0.00275  seconds. 

A  pulsation  of  the  impressed  e.m.f.,  e0,  of  a  frequency  higher 
than  250  cycles,  practically  cannot  penetrate  to  the  load  circuit, 
that  is,  does  not  appear  at  all  in  the  load  current  it  regardless 
of  how  much  a  pulsation  of  the  impressed  e.m.f.,  e0,  it  is,  and  a 


DIVIDED  CIRCUIT  139 

pulsation  of  impressed  e.m.f.,  e0,  of  a  frequency  of  120  cycles  re- 
appears in  the  load  current  iv  reduced  to  1  per  cent  of  its  value. 

In  cases  where  from  a  source  of  e.m.f.,  e0,  which  contains  a 
slight  high  frequency  pulsation  —  as  the  pulsation  corresponding 
to  the  commutator  segments  of  a  commutating  machine  —  a 
current  is  desired  showing  no  pulsation  whatever,  as  for  instance 
for  the  operation  of  a  telephone  exchange,  a  very  high  inductive 
reactance  in  series  with  the  circuit,  and  a  condensive  reactance 
in  shunt  therewith,  entirely  eliminates  all  high  frequency  pulsa- 
tions from  the  current,  passing  only  harmless  low  frequency 
pulsations  at  a  greatly  reduced  amplitude. 

81.  As  a  further  example  is  shown  in  Fig.  37  the  pulsation 
of  a  non-inductive  circuit,  xl  =  0,  of  the  resistance  rl  =  4  ohms, 
shunted  by  a  condensive  reactance  xc  =  10  ohms,  and  supplied 
over  a  line  of  resistance  r0  =  1  ohm  and  inductive  reactance 
x0  =  10  ohms,  by  an  impressed  e.m.f.,  e0  =  110  volts. 

Due  to  xl  =  0  equation  (58)  reduces  to 


1  0 

or,  substituting  numerical  values, 

a2  -  2.6  a  +  1.25  =  0 
and  at  ='0.637,          a2  =  1.963; 

that  is,  both  roots  are  real,  or  the  phenomenon  is  logarithmic. 
We  now  have 

l\-ty  +  A/-«-«7'+A4r'-B«', 

ia  =  -  0.255  A^-0'63™-  0.785  A^-1'"*'  , 

and  e  -  r,i,  =-  4  (i,'  +  Af-*'"1'  +  A/-1'963'). 

The  load  current  is 

i/  =  22  amp. 

A  reduction  of  the  impressed  e.m.f.,  e0,  by  10  per  cent,  or  from 
110  to  99  volts,  gives  the  integration  constants  Al  =  3.26  and 
A2  =  -  1.06;  hence, 

i,  =  19.8  +  3.26  e-0'637'  -  1.06  r1'**9  , 


and  e  =  4  ir 


140  TRANSIENT   PHENOMENA 

Without  a  condenser,  the  equation  of  current  would  be 
i  =  19.8  +  2.2  £-°-50. 

In  Fig.  37  is  shown,  with  6  as  abscissas,  the  drop  of  current 
il  and  i,  in  per  cent. 

Although  here  the  change  is  logarithmic,  while  in  the  former 
paragraph  it  was  trigonometric,  the  result  is  the  same  —  a  very 
great  reduction,  by  the  condenser,  of  the  drop  of  current  imme- 
diately after  the  change  of  e.m.f.  However,  in  the  present  case 


Fig.  37.  Suppression  of  pulsations  in  nori -inductive  direct-current  circuits  by 
series  inductance  and  shunted  capacity.  Effect  of  10  per  cent  drop  of 
voltage. 

the  change  of  the  circuit  is  far  more  rapid  than  in  the  preceding 
case,  due  to  the  far  lower  inductive  reactance  of  the  present  case. 
For  instance,  after  6  =  0.1,  the  drop  of  current,  with  condenser, 
is  0.045  per  cent,  without  condenser,  0.5  per  cent.  At  6  =  0.2, 
the  drop  of  current  is  0.23  and  0.95  per  cent  respectively.  For 
longer  times  or  larger  values  of  6,  the  difference  produced  by  the 
condenser  becomes  less  and  less. 

This  effect  of  a  condenser  across  a  direct-current  circuit,  of 
suppressing  high  frequency  pulsations  from  reaching  the  circuit, 
requires  a  very  large  capacity. 


CHAPTER  X. 

MUTUAL   INDUCTANCE. 

82.  In  the  preceding  chapters,  circuits  have  been  considered 
containing  resistance,  self-inductance,  and  capacity,  but  no 
mutual  inductance;  that  is,  the  phenomena  which  take  place 
in  the  circuit  have  been  assumed  as  depending  upon  the  impressed 
e.m.f.  and  the  constants  of  the  circuit,  but  not  upon  the 
phenomena  taking  place  in  any  other  circuit. 

Of  the  magnetic  flux  produced  by  the  current  in  a  circuit 
and  interlinked  with  this  circuit,  a  part  may  be  interlinked  with 
a  second  circuit  also,  and  so  by  its  change  generate  an  e.m.f.  in 
the  second  circuit,  and  part  of  the  magnetic  flux  produced  by 


Fig.  38.     Mutual  inductance  between  circuits. 

the  current  in  a  second  circuit  and  interlinked  with  the  second 
circuit  may  be  interlinked  also  with  the  first  circuit,  and  a 
change  of  current  in  the  second  circuit,  that  is,  a  change  of 
magnetic  flux  produced  by  the  current  in  the  second  circuit, 
then  generates  an  e.m.f.  in  the  first  circuit. 

Diagrammatically  the  mutual  inductance  between  two  circuits 
can  be  sketched  as  shown  by  M  in  Fig.  38,  by  two  coaxial  coils, 
while  the  self-inductance  is  shown  by  a  single  coil  L,  and  the 
resistance  by  a  zigzag  line. 

141 


142  TRANSIENT  PHENOMENA 

The  presence  of  mutual  inductance,  with  a  second  circuit, 
introduces  into  the  equation  of  the  circuit  a  term  depending 
upon  the  current  in  the  second  circuit. 

If  i^  =  the  current  in  the  circuit  and  r1  =  the  resistance  of 
the  circuit,  then  r^\  =  the  e.m.f.  consumed  by  the  resistance 
of  the  circuit.  If  L1  =  the  inductance  of  the  circuit,  that  is, 
total  number  of  interlinkages  between  the  circuit  and  the  number 
of  lines  of  magnetic  force  produced  by  unit  current  in  the  circuit, 
we  have 

di 

L{j±  =  e.m.f.  consumed  by  the  inductance, 
ctt 

where,  t  =  time. 

If  instead  of  time  t  an  angle  6  =  2  nft  is  introduced,  where  / 
is  some  standard  frequency,  as  60  cycles, 

di 

x.  3^  =  e.m.f.  consumed  by  the  inductance, 
au 

where  x1  =  2  nfL^  =  inductive  reactance. 

If  now  M  =  mutual  inductance  between  the  circuit  and 
another  circuit,  that  is,  number  of  interlinkages  of  the  circuit 
with  the  magnetic  flux  produced  by  unit  current  in  the  second 
circuit,  and  i2  =  the  current  in  the  second  circuit,  then 

M—~=  e.m.f.  consumed   by  mutual  inductance  in  the  first 
at 

circuit, 

M— -  =  e.m.f.  consumed  by  mutual  inductance  in  the  second 
at 

circuit. 

Introducing  xm  =  2  nfM  =  mutual  reactance  between  the 
two  circuits,  we  have 

di 

xm-^=  e.m.f.  consumed  by   mutual  inductance  in  the   first 
do 

circuit, 

di 

xm—±=  e.m.f.  consumed  by  mutual  inductance  in  the  second 
au 

circuit. 


MUTUAL  INDUCTANCE  143 

If  now  e^  =  the  e.m.f.  impressed  upon  the  first  circuit  and 
e2  =  the  e.m.f.  impressed  upon  the  second  circuit,  the  equations 
of  the  circuits  are 


dL  di7 

e,  =  r^  +  x^-±  +  xm  ^  +  xci 

and 


r 

J  il  dO  (1) 


-*2^+^^+*C2/V^,  (2) 

where  r1  =  the  resistance,  xl  =  2  7r/L1  =  the  inductive  re- 
actance, and  xci  =  =  the  condensive  reactance  of  the 
first  circuit;  r2  =  the  resistance,  x2  =  2  rfL2  =  the  inductive 
reactance,  xca  =  =  the  condensive  reactance  of  the 

second  circuit,  and  xm  =  2  nfM  =  mutual  inductive  reactance 
between  the  two  circuits. 

83.  In  these  equations,  xl  and  x2  are  the  total  inductive 
reactance,  Ll  and  L2  the  total  inductance  of  the  circuit,  that  is, 
the  number  of  magnetic  interlinkages  of  the  circuit  with  the 
total  flux  produced  by  unit  current  in  the  circuit,  the  self- 
inductive  flux  as  well  as  the  mutual  inductive  flux,  and  not 
merely  the  self-inductive  reactance  and  inductance  respectively. 

In  induction  apparatus,  such  as  transformers  and  induction 
machines,  it  is  usually  preferable  to  separate  the  total  reactance  z, 
into  the  self-inductive  reactance,  x81  referring  to  the  magnetic 
flux  interlinked  with  the  inducing  circuit  only,  but  with  no 
other  circuit,  and  the  mutual  inductive  reactance,  xm,  usually 
represented  as  a  susceptance,  which  refers  to  the  mutual  induc- 
tive component  of  the  total  inductance;  in  which  case 
x  =  xs  +  xm.  This  is  not  done  in  the  present  case. 

Furthermore  it  is  assumed  that  the  circuits  are  inductively 
related  to  each  other  symmetrically,  or  reduced  thereto;  that 
is,  where  the  mutual  inductance  is  due  to  coils  enclosed  in  the 
first  circuit,  interlinked  magnetically  with  coils  enclosed  in  the 
second  circuit,  as  the  primary  and  the  secondary  coils  of  a 
transformer,  or  a  shunt  and  a  series  field  winding  of  a  generator, 


144  TRANSIENT  PHENOMENA 

the  two  coils  are  assumed  as  of  the  same  number  of  turns,  or 
reduced  thereto. 

ri,       No.  turns  second  circuit 

If  a  =  —  =  —  =rr—        —  -  --  :  -  r—  ,  the  currents  in  the 
nA  No.  turns  first  circuit 

second  circuit  are  multiplied,  the  e.m.fs.  divided  by  a,  the  resis- 
tances and  reactances  divided  by  a2,  to  reduce  the  second  circuit 
to  the  first  circuit,  in  the  manner  customary  in  dealing  with 
transformers  and  especially  induction  machines.* 
If  the  ratio  of  the  number  of  turns  is  introduced  in  the  equa- 

tions, that  is,  in  the  first  equation  —  xm  substituted  for  xm,  in  the 

KI 

79 

second  equation  —  xm  for  xm,  and  the  equations  then  are 
n2 


and 


di~      n.      di. 


Since  the  solution  and  further  investigation  of  these  equations 
(3),  (4)  are  the  same  as  in  the  case  of  equations  (1)  and  (2),  except 
that  nl  and  n2  appear  as  factors,  it  is  preferable  to  eliminate  nl 
and  n2  by  reducing  one  circuit  to  the  other  by  the  ratio  of  turns 

M 

a  =  —  ,  and  then  use  the  simpler  equations  (1),  (2). 

ni 

84.  (A)  Circuits  containing  resistance,  inductance,  and  mutual 
inductance  but  no  capacity. 

In  such  a  circuit,  shown  diagrammatically  in  Fig.  38,  we  have 

di.  di~ 


~  * 

and  e2  =  r2i2  +  x2—  2  +  xm  ^  •  (6) 

Differentiating  (6)  gives 

de2  _      di2  d?i2  d2il 

~dd~~T2dd^  x*~dF~VXm^] 

*   See  the  chapters  on  induction  machines,  etc.,  in  "  Theory  and  Calcula- 
tion of  Alternating  Current  Phenomena." 


MUTUAL  INDUCTANCE  145 

from  (5)  follows 


/^  /  «  v*    *"~ 


, 
^        ~^r    "' 

and,  differentiated, 

fo 

* 


Substituting  (8)  and  (9)  in  (7)  gives 

del  de2  _         .  dil 

+  <*,*,-*-•>£,  (10) 

and  analogously, 

de2  de^ 


+  (*,*,  -  ^2)  ^  •  (ID 

Equations  (10)  and  (11)  are  the  two  differential  equations  of 
second  order,  of  currents  i\  and  iv 

If  e/,  i/  and  e/,  i2'  are  the  permanent  values  of  impressed 
e.m.fs.  and  of  currents  in  the  two  circuits,  and  e/',  if  and 
e2",  if  are  their  transient  terms,  we  have, 


e2  -  c/  +  el', 

Since  the  permanent  terms  must  fulfill  the  differential  equations 
(10)  and  (11), 

de'  de'  .  di/ 

W  +  x*^g-x'»W"=  Wl  +  (r'X2  +  T>X*}^9 
(Pi  ' 

+    (xiX2-Xm*)^  (12) 

and 

del          */  •  , 

r    +         "*"  ^      "  rr'  + 


(13) 


146  TRANSIENT  PHENOMENA 

subtracting  equations  (12)  and  (13)  from  (10)  and  (11)  gives 
the  differential  equations  of  the  transient  terms, 

de,"  def  x  di" 

Wi"  +  x*~dd-  ~  Xm^dd"'=  W"  +  (ri*3  +  Vi)  ~^T 

cPi  " 
+  (xA  -  xm>)  -^  (14) 

and 


*£ 

'*  dd      "m  dd       f1'2"2    '  V1~2  '  '2^  dd 

(Pi, 


2 
rle2    +  x1  -—      xm  ——-  —  rjT2i2    +  (r^  -f  T2Xj)  —  - 


(15) 


These  differential  equations  of  the  transient  terms  are  the 
same  as  the  general  differential  equations  (10)  and  (11)  and 
the  differential  equations  of  the  permanent  terms  (12)  and  (13). 

85.  If,  as  is  usually  the  case,  the  impressed  e.m.fs.  contain  no 
transient  term,  that  is,  the  transient  terms  of  current  do  not 
react  upon  the  sources  of  supply  of  the  impressed  e.m.fs.  and 
affect  them,  we  have 

e^  =  0    and  -  ea"  =  0; 

hence,  the  differential  equations  of  the  transient  terms  are 

di  (Pi 

0  =  rjj  +  (r,x2  +  r2x,)  —  +  (x,x2  -  xm2)  —          (16) 

and  are  the  same  for  both  currents  i"  and  i2",  that  is,  the 
transient  terms  of  currents  differ  only  by  their  integration 
constants,  or  the  terminal  conditions. 
Equation  (16)  is  integrated  by  the  function 

i  =  Ae-a9.  (17) 

Substituting  (17)  in  (16)  gives 

A£-a'{rir2  -  a  (r,x2  +  r2xj  +  a2  (x,x2  -  xm2)}  -  0; 

hence, 

A  —  indefinite,  as  integration  constant,  and 


_    r     2  '       rr      _r2 

*^W  *t/1*x/2  *°W 


MUTUAL  INDUCTANCE  147 

The  exponent  a  is  given  by  a  quadratic  equation  (18).  This 
quadratic  equation  (18)  always  has  two  real  roots,  and  in  this 
respect  differs  from  the  quadratic  equation  appearing  in  a  circuit 
containing  capacity,  which  latter  may  have  two  imaginary  roots 
and  so  give  rise  to  an  oscillation. 

Mutual  induction  in  the  absence  of  capacity  thus  always 
gives  a  logarithmic  transient  term;  thus, 


a  =  (r^2  +  T^}  \  ;  (T^      rf      +  *  T^Xm   •     (19) 

•>//y«/y»  /y»        *  \ 

Z/      ^jU/2  .t/fn    / 

As  seen,  the  term  under  the  radical  in  (19)  is  always  positive, 
that  is,  the  two  roots  al  and  a2  always  real  and  always  positive, 
since  the  square  root  is  smaller  than  the  term  outside  of  it. 

Herefrom  then  follows  the  integral  equation  of  one  of  the 
currents,  for  instance  iv  as 

o     --  <>'  '  4_   A   ff~aid4-  A   e-a*9  f90\ 

*i  T  »i     P  -Aj*  r  ^12£         i  \^) 

and  eliminating  from  the  two  equations  (5)  and  (6)  the  term 
dL 


pji  +  (x^x2  -  xm2)  -     +  xme2  -  x2e^    ,     (21) 


Tx 


2m 

leaving  the  two  integration  constants  Al  and  A2  to  be  deter- 
mined by  the  terminal  conditions,  as  0  =  0, 


and      i2  =  i2°. 


86.  If  the  impressed  e.m.fs.  e1  and  e2  are  constant,  we  have 


hence,  the  equations  of  the  permanent  terms  (12)  and  (13)  give 


thus: 


and 


^      and      tV  -  -  ;  (22) 

T  T 


(23) 


where,  A/  and  A2  follow  from  Al  and  A2  by  equation  (21). 


148  TRANSIENT  PHENOMENA 

If  the  mutual  inductance  between  the  two  circuits  is  perfect, 
that  is, 

xm2  =  x,xv  (24) 


-  r    2 

•^m 


equation  (18)  becomes,  by  multiplication  with  - 

r^2  -r  /  ^ 

a=rXr+\x  ;  (25) 

that  is,  only  one  transient  term  exists. 

As  example  may  be  considered  a  circuit  having  the  following 
constants:  et  ="  100  volts;  e2  =  0;  rt  =  5  ohms;  r2  =  5  ohms; 
xl  =  100  ohms;  x2  =  100  ohms,  and  xm  =  80  ohms.  This 
gives 

i'  =  20  amp.  and  i2   =  0, 

and 

a2  -  0.278  a  +  0.00695  =  0; 

the  roots  are       at  =  0.0278    and    a2  =  0.251 
and 


By  equation  (21), 

^=-25  +  1.25^  +  9^; 
hence, 


For  6  =  0  let  if  =  18  amp.,  or  the  current  10  per  cent  below 
the  normal,  and  if  =  0;  then  substituted,  gives: 

18  =  20  +  Al  +  A2    and    0  =  Al  -  A2, 
hence,  A^  =  A2  =  —  1; 

and  we  have 

il  =  20  —  (£-°-0278*  +  £-°'2510) 

and  i2=-  (e-°-0278  -£-°-251'). 


MUTUAL  INDUCTANCE  149 

87.  An  interesting  application  of  the  preceding  is  the  inves- 
tigation of  the  building  up  of  an  overcompounded  direct-current 
generator,  with  sudden  changes  of  load,  or  the  building  up,  or 
down,  of  a  compound  wound  direct-current  booster. 

While  it  would  be  desirable  that  a  generator  or  booster,  under 
sudden  changes  of  load,  should  instantly  adjust  its  voltage  to  the 
change  so  as  to  avoid  a  temporary  fluctuation  of  voltage,  actually 
an  appreciable  time  must  elapse. 

A  600-kw.  8-pole  direct-current  generator  overcompounds 
from  500  volts  at  no  load  to  600  volts  at  terminals  at  full  load 
of  1000  amperes.  The  circuit  constants  are:  resistance  of 
armature  winding,  r0  =  0.01  ohm;  resistance  of  series  field 
winding,  r2'  =  0.003  ohm;  number  of  turns  per  pole  in  shunt 
field  winding,  n1=  1000,  and  magnetic  flux  per  pole  at  500 
volts,  4>  =  10  megalines.  At  600  volts  full  load  terminal  voltage 
(or  voltage  from  brush  to  brush)  the  generated  e.m.f.  is  e  +  irQ 
=  610  volts. 

From  the  saturation  curve  or  magnetic  characteristics  of  the 
machine,  we  have: 

At  no  load  and  500  volts : 

5000  ampere-turns,  10  megalines  and  5  amp.  in  shunt  field 
circuit. 

At  no  load  and  600  volts : 

7000  ampere-turns  and  12  megalines. 

At  no  load  and  610  volts: 

7200  ampere-turns  and  12.2  megalines. 

At  full  load  and  600  volts: 

8500  ampere-turns,  12.2  megalines  and  6  amp.  in  shunt 
field. 

Hence  the  demagnetizing  force  of  the  armature,  due  to  the 
shift  of  brushes,  is  1300  ampere-turns  per  pole. 

At  600  volts  and  full  load  the  shunt  field  winding  takes 
6  amperes,  and  gives  6000  ampere-turns,  so  that  the  series  field 
winding  has  to  supply  2500  ampere-turns  per  pole,  of  which 
1300  are  consumed  by  the  armature  reaction  and  1200  magnetize. 

At  1000  amp.  full  load  the  series  field  winding  thus  has  2.5 
turns  per  pole,  of  which  1.3  neutralize  the  armature  reaction 
and  ft  2  —  1.2  turns  are  effective  magnetizing  turns. 


150  TRANSIENT  PHENOMENA 

The  ratio  of  effective  turns  in  series  field  winding  and  in  shunt 

field  winding  is  a  =  ^  =  1.2  X  10"3.    This  then  is  the  reduc- 
n, 

tion  factor  of  the  shunt  circuit  to  the  series  circuit. 

It  is  convenient  to  reduce  the  phenomena  taking  place  in  the 
shunt  field  winding  to  the  same  number  of  turns  as  the  series 
field  winding,  by  the  factors  a  and  a2  respectively. 

If  then  6  =  terminal  voltage  of  the  armature,  or  voltage 
impressed  upon  the  main  circuit  consisting  of  series  field  winding 
and  external  circuit,  the  same  voltage  is  impressed  upon  the 
shunt  field  winding  and  reduced  to  the  main  circuit  by  factor 
«,  gives  e,  =  ae  =  1.2  X  W~3  e. 

Since  at  500  volts  impressed  the  shunt  field  current  is  5 
amperes,  the  field  rheostat  must  be  set  so  as  to  give  to  the  shunt 

500 

field  circuit  the  total  resistance  of  r/  =  -  -  =  100  ohms. 

5 

Reduced  to  the  main  circuit  by  the  square  of  the  ratio  of 
turns,  this  gives  the  resistance, 

fi  =  av/  =  144  X  10~6  ohms. 

An  increase  of  ampere-turns  from  5000  to  7000,  corresponding 
to  an  increase  of  current  in  the  shunt  field  winding  by  2  amperes, 
increases  the  generated  e.m.f.  from  500  to  600  volts,  and  the 
magnetic  flux  from  10  to  12,  or  by  2  megalines  per  pole.  In 
the  induction  range  covered  by  the  overcompounding  from  500 
to  600  volts,  1  ampere  increase  in  the  shunt  field  increases  the 
flux  by  1  megaline  per  pole,  and  so,  with  nl  =  1000  turns,  gives 
109'  magnetic  interlinkages  per  pole,  or  8  X  109  interlinkages 
with  8  poles,  per  ampere,  hence  80  X  109  interlinkages  per  unit 
current  or  10  amperes,  that  is,  an  inductance  of  80  henrys. 
Reduced  to  the  main  circuit  this  gives  an  inductance  of  1.22  X 
10~6  X  80  =  115.2  X  10~6  henrys.  This  is  the  inductance  due 
to  the  magnetic  flux  in  the  field  poles,  which  interlinks  with 
shunt  and  series  coil,  or  the  mutual  inductance,  M  =  115.2  X 
10~6  henrys. 

Assuming  the  total  inductance  Lt  of  the  shunt  field  winding 
as  10  per  cent  higher  than  the  mutual  inductance  M,  that  is, 
assuming  10  per  cent  stray  flux,  we  have 

Lx  =  1.1  M  =  126.7  X  10~6  henrys. 


MUTUAL  INDUCTANCE  151 

In  the  main  circuit,  full  load  is  1000  amp.  at  600  volts.  This 
gives  the  effective  resistance  of  the  main  circuit  as  r  =  0.6  ohm. 

The  quantities  referring  to  the  main  circuit  may  be  denoted 
without  index. 

The  total  inductance  of  the  main  circuit  depends  upon  the 
character  of  the  load.  Assuming  an  average  railway  motor  load, 
the  inductance  may  be  estimated  as  about  L  =  2000  X  10~6 
henry s.  * 

In  the  present  problem  the  impressed  e.m.fs.  are  not  constant 
but  depend  upon  the  currents,  that  is,  the  sum  i  +  iv  where 
t\  =  shunt  field  current  reduced  to  the  main  circuit  by  the 
ratio  of  turns. 

The'  impressed  e.m.f.,  e,  is  approximately  proportional  to  the 
magnetic  flux  <£,  hence  less  than  proportional  to  the  current,  in 
consequence  of  magnetic  saturation.  Thus  we  have 

e  =  500  volts  for  5000  ampere-turns, 

5000 

or  i  +  il  —  — — -  =  4170  amp.  and 
\.2i 

e  =  600  volts  for  7200  ampere-turns, 

7200 

or  i  +  it  =  -    -  =  6000  amp. ; 
\.2i 

hence,  1830  amp.  produce  a 'rise  of  voltage  of  100,  or  1  amp. 
100         1 


raises  the  voltage  by 


1830      18.3 


6000 
At  6000  amp.  the  voltage  is  —    -  =  328  volts  higher  than  at 

lo.o 

0  amp.,  that  is,  the  voltage  in  the  range  of  saturation  between 
500  and  600  volts,  when  assuming  the  saturation  curve  in  this 
range  as  straight  line,  is  given  by  the  equation 


The  impressed  e.m.f.  of  the  shunt  field  is  the  same,  hence, 
reduced  to  the  main  circuit  by  the  ratio  of  turns,  a  =  1.2  X  10~3, 
is 


e,  = 


152 


TRANSIENT  PHENOMENA 


Assuming  now  as  standard  frequency,  /  =  60  cycles  per  sec., 
the  constants  of  the  two  mutually  inductive  circuits  shown 
diagrammatically  in  Fig.  38  are : 


Main  Circuit. 

Shunt  Field  Circuit. 

Current  

i  amp. 

il  amp 

Impressed  e.m.f... 
Resistance  

*=  272  1^  VOHS 

T  =    6  ohms 

e1  =  (272+~Ml.2X10-3volts 
rt  =  0  144X  10~3  ohms 

Inductance  
Reactance,  2xfL.  . 
Mutual  inductance 
Mutual  reactance  . 

L  =  2000X  ID"6  henrys 
x  =    755X  10-3  ohms 
M  =  115.2  > 
xm  =     43.5  > 

L1=  126.  7X10-6  henrys 
Xl=  47.8X10-3  ohms 
(  10-6  henrys 
;  10~3  ohms 

This  gives  the  differential  equations  of  the  problem  as 


and 


(26) 
(27) 


88.  Eliminating  —  from  equations  (26)  and  (27)  gives 
do 

di 

— l-  =  0.695  i  -  0.0712  ^  -  338. 

Equation  (28)  substituted  in  (26)  gives 

i   =  13.07  ~n  +  9.95  i  -  4950. 
ad 

Equation  (29)  substituted  in  (28)  gives 

J±  =    _  o.93  —  -  0.015  i  + 15. 
Equation  (29)  differentiated,  and  equated  with  (30),  gives 


+  0.828 


0.00115  i-  1.15  =  0. 


(28) 


(29) 


(30) 


(31) 


MUTUAL   INDUCTANCE  153 

Equation  (31)  is  integrated  by 

i  =  i'0  +  A£-°*. 

Substituting  this  in  (31)  gives 
Ara0{a?  -  0.828  a  +  0.00115}  +  {0.00115 iQ  -  1.15}  -=  0, 

hence,  i0  =  1000,  A  is  indefinite,  as  integration  constant,  and 
a2  -  0.828  a  +  0.00115  -  0; 

thus  a  =  0.414  ±  0.4126, 

and  the  roots  are 

at  =  0.0014    and    a2  =  0.827. 

Therefore 

i  =  1000  +  A/-0-0014'  +  A/"0'827'.  (32) 

Substituting  (32)  in  (29)  gives 

i,  -  5000  +  9.932  A/"0-0014'-  0.85  A2r  °'827'.          (33) 

Substituting  in  (32)  and  (33)  the  terminal  conditions  0  =  0, 
i  =  0,  and  il  =  4170,  gives 

At  +  A2  =  --  1000    and    9.932  At  -  0.85  A2  =     -  830, 

that  is, 

At  =     -  156    and    A2  =    -  844. 

Therefore 

i  =  1000  -  156  e-°'™9  -  844  r0'827'  (34) 

and 

i\  =  5000  -  1550  r  °-00140  +  720  r  °'827fl ;  (35) 

or  the  shunt  field  current  t\  reduced  back  to  the  number  of  turns 
of  the  shunt  field  by  the  factor  a  «  1.2  X  10~3  is 

i/  =  6  -  1.86  r0'™140  +  0.86  r0'827,  (36) 


154  TRANSIENT  PHENOMENA 

and  the  terminal  voltage  of  the  machine  is 
e 


272+^, 


18.3 


—  0.8270 


or>  e  =  600  -  93.2  e~{          -  6.8  £~l   "y.  (37) 

As  seen,  of  the  two  exponential  terms  one  disappears  very 
quickly,  the  other  very  slowly. 

Introducing  now  instead  of  the  angle  6  =  2  nft  the  time,  t, 
gives  the  main  current  as 

i  =  1000  -  156  £-°'53'  -  844  r*1" 


the  shunt  field  current  as 


i>  =  6-  1.86  £-°-53<  +  0.86 


and  the  terminal  voltage  as 

e  =  600  -  93.2  £-°'53<  -  6.8  e- 


(38) 


89.   Fig.  39  shows  these  three  quantities,  with  the  time,  t,  as 
abscissas. 


1=0.010.02 


i  6 

900  4.8  f 

800  4.6 

700  4.4 

600  4.2] 

500  6. 

400  5.8 1 

300  5.6 

200  5.4 

100  5.2] 

1000  5. 

900  4.81 

800  4.6 

700  4.4 
600  L. 

4.o 


=  Termina 


i  --MUncn 


Seconds 
0.04       0.05       0.06      0.07      0.08 


voltiige 


currei 


3  ohms 
at. 
ohms 
^ 

-96-mh 


Mr 


t  =-12345678 
Seconds 

Fig.  39.     Building-up  of  over-compounded  direct-current  generator  from 
600  volts  no  load  to  600  volts  load. 

The  upper  part  of  Fig.  39  shows  the  first  part  of  the  curve 
with  100  times  the  scale  of  abscissas  as  the  lower  part.  As  seen, 
the  transient  phenomenon  consists  of  two  distinctly  different 


MUTUAL  INDUCTANCE  155 

periods:  first  a  very  rapid  change  covering  a  part  of  the  range 
of  current  or  e.m.f.,  and  then  a  very  gradual  adjustment  to  the 
final  condition. 

So  the  main  current  rises  from  zero  to  800  amp.  in  0.01  sec., 
but  requires  for  the  next  100  amp.,  or  to  rise  to  a  total  of  900 
amp.,  about  a  second,  reaching  95  per  cent  of  full  value  in  2.25 
sec.  During  this  time  the  shunt  field  current  first  falls  very 
rapidly,  from  5  amp.  at  start  to  4.2  amp.  in  0.01  sec.,  and  then, 
after  a  minimum  of  4.16  amp.,  at  t  =  0.015,  gradually  and  very 
slowly  rises,  reaching  5  amp.,  or  its  starting  point,  again  after 
somewhat  more  than  a  second.  After  2.5  sec.  the  shunt  field 
current  has  completed  half  of  its  change,  and  after  5.5  sec.  90 
per  cent  of  its  change. 

The  terminal  voltage  first  rises  quickly  by  a  few  volts,  and 
then  rises  slowly,  completing  50  per  cent  of  its  change  in  1.2 
sec.,  90  per  cent  in  4.5  sec.,  and  95  per  cent  in  5.5  sec. 

Physically,  this  means  that  the  terminal  voltage  of  the  machine 
rises  very  slowly,  requiring  several  seconds  to  approach  station- 
ary conditions.  First,  the  main  current  rises  very  rapidly,  at  a 
rate  depending  upon  the  inductance  of  the  external  circuit,  to 
the  value  corresponding  to  the  resistance  of  the  external  circuit 
and  the  initial  or  no  load  terminal  voltage,  and  during  this 
period  of  about  0.01  sec.  the  magnetizing  action  of  the  main 
current  is  neutralized  by  a  rapid  drop  of  the  shunt  field  current. 
Then  gradually  the  terminal  voltage  of  the  machine  builds  up, 
and  the  shunt  field  current  recovers  to  its  initial  value  in  1.15 
sec.,  and  then  rises,  together  with  the  main  current,  in  corre- 
spondence with  the  rising  terminal  voltage  of  the  machine. 

It  is  interesting  to  note,  however,  that  a  very  appreciable 
time  elapses  before  approximately  constant  conditions  are 
reached. 

90.  In  the  preceding  example,  as  well  as  in  the  discussion  of 
the  building  up  of  shunt  or  series  generators  in  Chapter  II,  the 
e.m.fs.  and  thus  currents  produced  in  the  iron  of  the  magnetic 
field  by  the  change  of  the  field  magnetization  have  not  been 
considered.  The  results  therefore  directly  apply  to  a  machine 
with  laminated  field,  but  only  approximately  to  one  with  solid 
iron  poles. 

In  machines  with  solid  iron  in  the  magnetic  circuit,  currents 
produced  in  the  iron  act  as  a  second  electric  circuit  in  inductive 


156  TRANSIENT  PHENOMENA 

relation  to  the  field  exciting  circuit,  and  the  transition  period 
thus  is  slower. 

As  example  may  be  considered  the  excitation  of  a  series 
booster  with  solid  and  with  laminated  poles;  that  is,  a  machine 
with  series  field  winding,  inserted  in  the  main  circuit  of  a  feeder, 
for  the  purpose  of  introducing  into  the  circuit  a  voltage  propor- 
tional to  the  load,  and  thus  to  compensate  for  the  increasing 
drop  of  voltage  with  increase  of  load. 

Due  to  the  production  of  eddy  currents  in  the  solid  iron  of  the 
field  magnetic  circuit,  the  magnetic  flux  density  is  not  uniform 
throughout  the  whole  field  section  during  a  change  of  the  mag- 
netic field,  since  the  outer  shell  of  the  field  iron  is  magnetized  by 
the  field  coil  only,  while  the  central  part  of  the  iron  is  acted  upon 
by  the  impressed  m.m.f.  of  the  field  coil  and  the  m.m.f.  of  the 
eddy  currents  in  the  outer  part  of  the  iron,  and  the  change  of 
magnetic  flux  density  in  the  interior  thus  lags  behind  that  of 
the  outside  of  the  iron.  As  result  hereof  the  eddy  currents  in 
the  different  layers  of  the  structure  differ  in  intensity  and  in 
phase. 

A  complete  investigation  of  the  distribution  of  magnetism  in 
this  case  leads  to  a  transient  phenom- 
enon in  space,  and  is  discussed  'in 
Section  III.  For  the  present  purpose, 
where  the  total  m.m.f.  of  the  eddy 
currents  is  small  compared  with  that 
of  the  main  field,  we  can  approxi- 
mate the  effect  of  eddy  currents  in 
the  iron  by  a  closed  circuit  second- 
ary conductor,  that  is,  can  assume 
uniform  intensity  and  phase  of 
secondary  currents  in  an  outer  layer  Fig'  40'  tSection  °f  a  mag' 

,,  , ,      .    J      , ,     ,  .  .,       ,,  netic  circuit. 

of  the  iron,  that  is,  consider  the  outer 

layer  of  the  iron,  up  to  a  certain  depth,  as  a  closed  circuit 

secondary. 

Let  Fig.  40  represent  a  section  of  the  magnetic  circuit  of  the 
machine,  and  assume  uniform  flux  density.  If  <£  =  the  total 
magnetic  flux,  lr=  the  radius  of  the  field  section,  then  at  a 
distance  I  from  the  center,  the  magnetic  flux  enclosed  by  a 

/  l\2 
circle  with  radius  I  is  f  — J  4>,  and  the  e.m.f.  generated  in  the 


zone 


MUTUAL  IXDUCTANCE  157 

/  l\2 
at  distance  I  from  the  center  is  proportional  to  f— J  4>,that 

/l\2 
is,e  =  a  (—  j  4>.     The  current  density  of  the  eddy  currents  in 

this  zone,  which  has  the  length  2  Til,  is  therefore  proportional  to 

e  bl 

— - ,  or  is  i  =  —  <£.    This  current  density  acts  as  a  m.m.f .  upon 

2i    Til  lr 

/l\2 
the  space  enclosed  by  it,  that  is,  upon  f— j    of  the  total  field 

section,  and  the  magnetic  reaction  of  the  secondary  current  at 

/l\2 

distance  I  from  the  center  therefore  is  proportional  to  i  f — j  ,  or 

vr/  x 

is  &  =  —  $,  and  therefore  the  total  magnetic  reaction  of  the 
eddy  currents  is 


At  the  outer  periphery  of  the  field  iron,  the  generated  e.m.f. 

is    et  =  a<£,  the  current  density  therefore  il  =  y<f»;    and    the 

ir 

f* 

magnetic  reaction  $  ^  =  —  <l>,  and  therefore 

lr 


that  is,  the  magnetic  reaction  of  the  eddy  currents,  assuming 
uniform  flux  density  in  the  field  poles,  is  the  same  as  that  of  the 

currents  produced  in  a  closed  circuit  of  a  thickness  -j,  or  one- 

fourth  the  depth  of  the  pole  iron,  of  the  material  of  the  field  pole 
and  surrounch'ng  the  field  pole,  that  is,  fully  induced  and  fully 
magnetizing. 
The  eddy  currents  in  the  solid  material  of  the  field  poles  thus 

can  be  represented  by  a  closed  secondary  circuit  of  depth  -^ 

surrounding  the  field  poles. 

The  magnitude  of  the  depth  of  the  field  copper  on  the  spools 


158  TRANS  I  EN  T  PHENOMENA 

is  probably  about  one-fourth  the  depth  of  the  field  poles.  Assum- 
ing then  the  width  of  the  band  of  iron  which  represents  the 
eddy  current  circuit  as  about  twice  the  width  of  the  field  coils 
—  since  eddy  currents  are  produced  also  in  the  yoke  of  the 
machine,  etc.  —  and  the  conductivity  of  the  iron  as  about  0.1 
that  of  the  field  copper,  the  effective  resistance  of  the  eddy 
current  circuit,  reduced  to  the  field  circuit,  approximates  five 
times  that  of  the  field  circuit. 

Hence,  if  r2  —  resistance  of  main  field  winding,  rl  =  5  r2  = 
resistance  of  the  secondary  short  circuit  which  represents  the 
eddy  currents. 

Since  the  eddy  currents  extend  beyond  the  space  covered  by 
the  field  poles,  and  considerably  down  into  the  iron,  the  self- 
inductance  of  the  eddy  current  circuit  is  considerably  greater 
than  its  mutual  inductance  with  the  main  field  circuit,  and  thus 
may  be  assumed  as  twice  the  latter. 

91.  As  example,  consider  a  20Okw.  series  booster  covering 
the  range  of  voltage  from  0  to  200,  that  is,  giving  a  full  load 
value  of  1000  amperes  at  200  volts.  Making  the  assumptions 
set  forth  in  the  preceding  paragraph,  the  following  constants 
are  taken:  the  armature  resistance  =  0.008  ohms  and  the 
series  field 'winding  resistance^  0.004  ohm;  hence,  the  short 
circuit  —  or  eddy  current  resistance  —  rl  =  0.02  ohm.  Further- 
more let  M  =  900  X  10~6  henry  =  mutual  inductance  between 
main  field  and  short-circuited  secondary;  hence,  xm  =  0.34  ohm 
=  mutual  reactance,  and  therefore,  assuming  a  leakage  flux  of 
the  secondary  equal  to  the  main  flux,  Ll  =  1800  X  10~6  henry 
and  xl  =  0.68  ohm. 

The  booster  is  inserted  into  a  constant  potential  circuit  of  550 
volts,  so  as  to  raise  the  voltage  from  550  volts  no  load  to  750 
volts  at  1000  amperes. 

The  total  resistance  of  the  circuit  at  full  load,  including  main 
circuit  and  booster,  therefore  is  r  =  0.75  ohm. 

The  inductance  of  the  external  circuit  may  be  assumed  as 
L  =  4500  X  10~6  henrys;  hence,  the  reactance  at/  =  60  cycles 
per  sec.  is  x  =  1.7  ohms.  The  impressed  e.m.f.  of  the  circuit  is 
e  =  550  +  e',  ef  being  the  e.m.f.  generated  in  the  booster. 
Since  at  no  load,  for  i  =  0,  e7  =  0,  and  at  full  load,  for  i  =  1000, 
e'  =  200,  assuming  a  straight  line  magnetic  characteristic  or 
saturation  curve,  that  is,  assuming  the  effect  of  magnetic  satura- 


MUTUAL  INDUCTANCE 


159 


tion  as  negligible  within  the  working  range  of  the  booster,  we 
have 

e  =  550  +  0.2  (i  +  ij. 

This  gives  the  following  constants : 


Main  Circuit. 

Eddy  Current  Circuit. 

Current  
Impressed  e.m.f  
Resistance  
Inductance  
Reactance 

i  amp. 
e=  550+0.2(i+t,)  volts. 
r=0.75  ohm. 
L=4500X10~flhenrys. 
x—  I  7  ohms 

il  amp. 
0  volts. 
r,  =  0.02  ohm. 
L1=1800X10-«henrys. 
xl  —  0  68  ohm 

Mutual  inductance.  .  . 
Mutual  reactance  .... 

M=  900  : 
xm  =  0.34  ( 

<  10""6  henrys. 
mm. 

This  gives  the  differential  equations  of  the  problem  as 


and 


550  -  0.55  i  +  0.2  L  -  1.7       -  0.34        =  0 

da  do 


0.02^  +  0.34^-  +  0.68^  =  0* 
ad  ad 


Adding  2  times  (39)  to  (40)  gives 


di 


1100  -  1.1  i  +  0.42  i.  -  3.06  —  =  0, 

dv 

or  t\  =  7.28  ^  +  2.62  i  -  2620, 

herefrom:      0.02  L  =  0.1456  4  +  0.0524  i  -  52.4, 

ay 


and 


^  =  4.95^1.781, 


substituting  the  last  two  equations  into  (40), 


+  0.458        +  0.0106  i  -  10.6  =  0. 
ad  du 


If 
then 


i  =  in  +  Ae 


-aff 


(39) 
(40) 

(41) 
(42) 
(43) 
(44) 

(45) 
(46) 


At-<*(c?  -  0.45S  a  +  0.0106)  +  0.0106  it  -  10.6  =  0. 


160 


TRANSIENT  PHENOMENA 


As  transient  and  permanent  terms  must  each  equal  zero, 
i0  =  1000    and    a2  -  0.458  a  +  0.0106  =  0, 

wherefrom  a  =  0.229  ±  0.205; 

the  roots  are         o4  =  0.024    and    a2  =  0.434; 

then  we  have 


i  =  1000  +  A<e-°-™6  + 


-0.4340 


and 


2.45  A  1( 


-0.0240 


-  0.55  A2e 


-0.4340 


and 


With  terminal  conditions  0  =  0,  i  =  0,  and  il  =  0, 
Al  =     -  183    and    A2  =  -  817. 

It6  =  2xft  =  377.5  J,  we  have 

i  =  1000  -  183  e-™71-  817  e~mt, 
^  =  -  450  {e-«.o7«  _  £-i64^ 

e  =  750  -127e-9-07<-  73~164<. 


0.01        0.02        0.03         0.04        O.Oo         0.06        O.OT        0.08         0.09        0.10 


(47) 
(48) 


(49) 


Fig.  41.     Building  up  of  feeder  voltage  by  series  booster. 

In  the  absence  of  a  secondary  circuit,  or  with  laminated  field 
poles,  equation  (39)  would  assume  the  form  it  =  0,  or 


550  +  0.2  1  -  0.75i 


1.7—; 
ad 


(50) 


hence, 

and 

or 
and 


—  =  0.323  (1000  -  i) 
ad 


i  =  1000  (1  - 


—  0.323  0  ' 


i  =  1000(1  -  e-122')! 
e  =  750  -  200£-122<;J 


(51) 


MUTUAL  INDUCTANCE  161 

that  is,  the  e.m.f.,  e,  approaches  final  conditions  at  a  more  rapid 
rate. 

Fig.  41  shows  the  curves  of  the  e.m.f.,  e,  for  the  two  conditions, 
namely,  solid  field  poles,  (49),  and  laminated  field  poles,  (51). 

(B)  Mutual  inductance  in  circuits  containing  self-inductance 
and  capacity. 

92.  The  general  equations  of  such  a  pair  of  circuits,  (3)  and 
(4),  differentiated  to  eliminate  the  integral  give 

^l  =  x  i  +  r^l  +  x?il  +  Xm^  (52) 

and 

and  the  potential  differences  at  the  condensers,  from  (3)  and  (4), 
are 

and 

(*  di,  di. 

,    £  /v.  |  .  /Ek^i) 

'2~JZ  ~   Xm   Jn  {°°) 


If  now  the  impressed  e.m.fs.,  el  and  ev  contain  no  transient 
term,  that  is,  if  the  transient  values  of  currents  il  and  i2  exert 
no  appreciable  reaction  on  the  source  of  e.m.f.,  and  if  {/  and  i2 
are  the  permanent  terms  of  current,  then,  substituting  t/  and 
i2'  in  equations  (52)  and  .(53),  and  subtracting  the  result  of  this 
substitution  from  (52)  and  (53),  gives  the  equations  of  the 
transient  terms  of  the  currents  \  and  i2,  thus : 

di.  tfi.  d?i2 

^  =          +  r-     +  xl-+xm—  (56) 


and 


di2          d?i2  cPil 

+  x—  +  xm  —  >  (57) 


de  dc~ 

If  the  impressed  e.m.fs.,  et  and  e2,  are  constant,  -r-  and  — 


162  TRANSIENT  PHENOMENA 

equal  zero,  and  equations  (52)  and  (53)  assume  the  form  (56) 
and  (57);  that  is,  equations  (56)  and  (57)  are  the  differential 
equations  of  the  transient  terms,  for  -the  general  case  of  any 
e.m.fs.,  e^  and  e2,  which  have  no  transient  terms,  and  are  the 
general  differential  equations  of  the  case  of  constant  impressed 
e.m.fs.,  el  and  er 
From  (56)  it  follows  that 

d?i2  .  dil          (Pi^ 

Xm~d(P  ~~     ~  Xcih  ~  TI  ~dd  ~  XlW2  ' 

Differentiating  equation  (57)  twice,  and  substituting  therein 
(58),  gives 


2X  d4i      ,  N  (Pi       , 

-  xm2)  —  +  (r,x2  +  r2xj  —  +  (xcix2 


,2       2  ci2        C2,        ta 

+  (  Vi  +  V  J  ^  +  xcxC2  i  =  0.  (59) 

This  is  a  differential  equation  of  fourth  order,  symmetrical  in 
r^x^  and  r2x2xC2,  which  therefore  applies  to  both  currents, 
il  and  ir 

The  expressions  of  the  two  currents  il  and  i2  therefore  differ 
only  by  their  integration  constants,  as  determined  by  the  ter- 
minal conditions. 

Equation  (59)  is  integrated  by 

i  =  As"*  (60) 

and  substituting  (60)  in  (59)  gives  for  the  determination  of  the 
exponent  a  the  quartic  equation 

(x&  -  xm2)  a4  -  (r^2  +  r2xj  a3  +  (xcix2 

-  (xcir2  +  x^)  a  +  xClxC2  =  0, 
or 


2^1  ~3     i       XCt 

a  -\-  • 


,x2  -  xm2 


The  solution  of  this  quartic  equation  gives  four  values  of  a, 
and  thus  gives 

*-**9.          (62) 


MUTUAL    INDUCTANCE  163 

The  roots,  a,  may  be  real,  or  two  real  and  two  imaginary,  or 
all  imaginary,  and  the  solution  of  the  equation  by  approxima- 
tion therefore  is  difficult. 

In  the  most  important  case,  where  the  resistance,  r,  is  small 
compared  with  the  reactances  x  and  xc  —  and  which  is  the  only 
case  where  the  transient  terms  are  prominent  in  intensity  and 
duration,  and  therefore  of  interest  —  as  in  the  transformer  and 
the  induction  coil  or  Ruhmkorff  coil,  the  equation  (61)  can  be 
solved  by  a  simple  approximation. 

In  this  case,  the  roots,  a,  are  two  pairs  of  conjugate  imaginary 
numbers,  and  the  phenomenon  oscillatory. 

The  real  components  of  the  roots,  a,  must  be  positive,  since 
the  exponential  s~a9  must  decrease  with  increasing  0. 

The  four  roots  thus  can  be  written : 


(63) 


where  a  and  /?  are  positive  numbers. 

In  the  equation  (61),  the  coefficients  of  a3  and  a  are  small, 
since  they  contain  the  resistances  as  factor,  and  this  equation 
thus  can  be  approximated  by 


a4  +    e  <*a2  +       c       2  =  0;  (64) 

x 


hence, 

'  xcix2 


_ 


that  is,  a2  is  negative,  having  two  roots, 

6t  =    -  /V    and    6,  -  -  /?/. 
This  gives  the  four  imaginary  roots  of  a  as  first  approximation  : 

° 


164  TRANSIENT  PHENOMENA 

If  av  a2,  ag,  a4  are  the  four  roots  of  equation  (61),  this  equation 
can  be  written 

/(a)  =  ( a-  oj  (a  -  aa)  (a  -  a3)  (a  -  a4)  -  0; 
or,  substituting  (63), 

/(a)  =  {(a  -  a,Y  +  ft'}  {(a  -  «2)2  +  ft2}  =  0,     (67) 
and  comparing  (67)  with  (61)  gives  as  coefficients  of  a3  and  of  a, 


2  (a,  4-  «2)  =  ? 


and 


(68) 


and  since  /?j2  and  /?22  are  given  by  (65)  and  (66)  as  roots  of  equa- 
tion (64),  av  a  2,  fv  /?2,  and  hereby  the  four  roots  av  a2,  a3,  a4  of 
equation  (61)  are  approximated  by  (64),  (65),  (66),  (68). 

The  integration  constants  Av  A2,  A3,  A4  now  follow  from  the 
terminal  conditions. 

93.  As  an  example  may  be  considered  the  operation  of  an 
inductorium,  or  Ruhmkorff  coil,  by  make  and  break  of  a  direct- 
current  battery  circuit,  with  a  condenser  shunting  the  break,  in 
the  usual  manner. 

Let  el  =  10  volts  =  impressed  e.m.f.;  r1  =  0.4  ohm  = 
resistance  of  primary  circuit,  giving  a  current,  at  closed  circuit 
and  in  stationary  condition,  of  i0=  25  amp.;  r2=  0.2  ohm  = 
resistance  of  secondary  circuit,  reduced  to  the  primary  by  the 
square  of  the  ratio  of  primary  -=-  secondary  turns ;  xl  =  10  ohms 
=  primary  inductive  reactance;  x2  =  10  ohms  =  secondary 
inductive  reactance,  reduced  to  primary;  xm  =  8  ohms  =  mutual 
inductive  reactance;  xCi  =  4000  ohms  =  primary  condensive 
reactance  of  the  condenser  shunting  the  break  of  the  interrupter 
in  the  battery  circuit,  and  xC2  =  6000  ohms  =  secondary 
condensive  reactance,  due  to  the  capacity  of  the  terminals  and 
the  high  tension  winding. 

Substituting  these  values,  we  have 

BI  =  10  volts  i0  =  25  amp. 

rt  =  0.4  ohm        xl  =  10  ohms        xCi  =  4000  ohms 
r2  =  0.2  ohm        x2  =  10  ohms        xC2  =  6000  ohms 
xm  =  8  ohms. 


(69) 


MUTUAL  INDUCTANCE 


165 


These  values  in  equation  (61)  give 

/  (a)  =  a4-  0.167  a3  +  2780  a2  -  89  a  +  667,000  =  0,     (70) 
and  in  equation  (64)  they  give 

ft  (a)  =  a4  +  2780  a2  +  667,000  =  0 


and 

or 

hence, 

and 


=-(1390  ±1125) 
=  -2515, 
=  -265; 
ft  =  50.15 
=  16.28. 


From  (68)  it  follows  that 

ai  +  a2  =  0.0833 

and  265  a^  +  2515  a2  =  44.5; 

hence,  al  -  0.073, 

a2  =  0.010. 

Introducing  for  the  exponentials  with  imaginary  exponents  the 
trigonometric  functions  give 

.!  cos  50.15  d  +  A2  sin  50.15  d } 

sin  16.280 

(71) 
^  cos  50.150  +  C2  sin  50.150} 

>!  cos  16.28  6  +  D2  sin  16.28  0 

where  the  constants  C  and  D  depend  upon  A  and  B  by  equations 

(56),  (57),  or  (58),  thus: 
Substituting  (71)  into  (58), 


=0 


(58) 


gives  an  identity,  from  which,  by  equating  the  coefficients  of 
£-°*  cos  bd  and  e.'00  sin  bd  to  zero,  result  four  equations,  in  the 
coefficients 

A,  B,  C,  D, 

A,  B,  C,  D2, 


166  TRANSIENT  PHENOMENA 

from  which  follows,  with  sufficient  approximation, 

A!  =    -0.95^ 
A2  =    -0.98(72 
Bl  =   +  1.57  A 
£2=   +  1.57  A; 
hence, 

\  =  -  0.96  £-0-073^  ci  cos  50.15  0  -f  C2  sin  50.15 
+  1.57  £-™io0  j  A  cos  16.28  0  +  D2  sin  16.28 

and  substituting  (71)  and  (73)  in  the  equations  of  the  condenser 
potential,  (54)  and  (55),  gives 

e,f  =  10  +  79  s~OJm9{C2  cos  50.15 0  -  Ct  sin  50.15 0 


(72) 


-  385  £-°-010'{  Z)2  cos  16.28  0  -  Dl  sin  16.28  0 


(74) 


e2'  =  118  £-°-™e{C2  cos  50.15  0  -  (7X  sin  50.15  0} 

+  367£-°-010'{A  cos  16.280  -  A  sin  16.280} 

94.    Substituting  now  the  terminal  conditions  of  the  circuit : 
At  the  moment  where  the  interrupter  opens  the  primary 

circuit  the  current  in  this  circuit  is   \  =  --  -  =  25  amp.     The 

condenser  in  the  primary  circuit,  which  is  shunted  across  the 
break,  was  short-circuited  before  the  break,  hence  of  zero  poten- 
tial difference.     The   secondary  circuit  was  dead.     This  then 
gives  the  conditions  0  =  0;^  =  25,  ?'2  =  0,  e^  =  0,  and  e/  =  0. 
Substituting  these  values  in  equations  (71),  (73),  (74)  gives 

25=    -0.95    C,  +  1.58  A 
0  =  '  Ci  +         A 


0  =  10  +  79  Ca  -  385  D2 


=  118(7, 


+367ZX 


hence 


C,=    -  10 

(72  =    -  0.05  =*  0 

A  =    +  10 

0.016  =*  0, 


D2  = 


MUTUAL  INDUCTANCE  167 

and 
i,  =  9.6  s-0™9  cos  50.15  0  +  15.7  £-°Moe  cos  16.28  6 

(75) 


i2  =  -10  £-°073'  cos  50.15  (9  +  10  £-°-010'  cos  16.28  0 
e/  =  10  +  790  £-°-073'  sin  50.15  6  +  3850  £-°-010'  -sin  16.28  6 


Approximately  therefore  we  have 

i\  =  9.6  £-°-073e  cos  50.15  0  +  15.7  £-°-010'  cos  16.28  0 
ia  =    - 10  { £-°-0730  cos  50.15  0  -  £-°mo0  cos  16.28  0  } 

e/  =  3850  fi-OJMO«  sin  16.28  0 

<  -   -3670  £-°010'  sin  16.28  6. 

The  two  frequencies  of  oscillation  are  3009  and  977  cycles 
per  sec.,  hence  rather  low. 

The  secondary  terminal  voltage  has  a  maximum  of  nearly 
4000,  reduced  to  the  primary,  or  400  times  as  large  as  corre- 
sponds to  the  ratio  of  turns. 

In  this  particular  instance,  the  frequency  3009  is  nearly 
suppressed,  and  the  main  oscillation  is  of  the  frequency  977. 


CHAPTER  XL 

GENERAL  SYSTEM  OF  CIRCUITS. 

(A)  Circuits  containing  resistance  and  inductance  only. 

95.  Let,  upon  a  general  system  or  network  of  circuits  con- 
nected with  each  other  directly  or  inductively,  and  containing 
resistance  and  inductance,  but  no  capacity,  a  system  of  e.m.fs., 
ey  be  impressed.  These  e.m.fs.  may  be  of  any  frequency  or 
wave  shape,  or  may  be  continuous  or  anything  else,  but  are 
supposed  to  be  given  by  their  equations.  They  may  be  free  of 
transient  terms,  or  may  contain  transient  terms  depending  upon 
the  currents  in  the  system.  In  the  latter  case,  the  dependency 
of  the  e.m.f.  upon  the  currents  must  obviously  be  given. 

Then,  in  each  branch  circuit, 

^5~=0,  (1) 

where  e  =  total  impressed  e.m.f.;  r.  =  resistance;  L  =  induc- 
tance, of  the  circuit  or  branch  of  circuit  traversed  by  current  i, 
and  Ms  =  mutual  inductance  of  this  circuit  with  any  circuit  in 
inductive  relation  thereto  and  traversed  by  current  is. 

The  currents  in  the  different  branch  circuits  of  the  system 
depend  upon  each  other  by  Kirchhoff's  law, 

D  i  -  0  (2) 

at  every  branching  point  of  the  system. 

By  equation  (2)  many  of  the  currents  can  be  eliminated  by 
expressing  them  in  terms  of  the  other  currents,  but  a  certain 
number  of  independent  currents  are  left. 

Let  n  =  the  number  of  independent  currents,  denoting  these 
currents  by  i-K,  where  K  =  1,  2,  .  . .  n.  (3) 

Usually,  from  physical  considerations,  the  number  of  inde- 
pendent currents  of  the  system,  n,  can  immediately  be  given. 

168 


GENERAL  SYSTEM  OF  CIRCUITS  169 

For  these  n  currents  iK,  n  independent  differential  equations 
of  form  (1)  can  be  written  down,  between  the  impressed  e.m.fs. 
ey  or  their  combinations,  and  currents  which  are  expressed  by 
the  n  independent  currents  iK.  They  are  given  by  applying 
equation  (1)  to  a  closed  circuit  or  ring  in  the  system. 

These  equations  are  of  the  form 

eq~r  V'*~?K  c^  =  0  |  (4) 

where  q  =  1,  2, ...  n, 

where  the  n2  coefficients  bq  are  of  the  dimension  of  resistance )  ,-\ 
and  the  n2  coefficients  cKq  of  the  dimension  of  inductance.        )  ^ 

These  n  simultaneous  differential  equations  of  n  variables  iK 
are  integrated  by  the  equations 


'    A?,-*,  (6) 

1 

where  iKf  is  the  stationary  value  of  current  iK,  reached  for  t  =  <x>  . 
Substituting  (6)  in  (4)  gives 


M^-^O.  (7) 

1 


For  t  =  oo  ,  this  equation  becomes 


These  n  equations  (8)  determine  the  stationary  components 
of  the  n  currents,  iK'. 

Subtracting  (8)  from  (7)  gives,  for  the  transient  components 
of  currents  iK, 


the  n  equations 


170  TRANSIENT  PHENOMENA 

Reversing  the  order  of  summation  in  (10)  gives 

A-o        =0- 


(11) 


The  n  equations  (11)  must  be  identities,  that  is,  the  coefficients 
of  £~aJ  must  individually  disappear.  Each  equation  (11)  thus 
gives  m  equations  between  the  constants  a,  A,  b,  c,  for  i  =  1, 
2,  .  .  .  m,  and  since  n  equations  (11)  exist,  we  get  altogether  mn 
equations  of  the  form 


where 


-0, 


q  =  1,  2,  3,.  .  .  n    and    i  =  1,  2,  3,.  .  .  m. 


(12) 


In  addition  hereto,  the  n  terminal  conditions,  or  values  of 
current  iK"  for  t  =  0:  iK°,  give  by  substitution  in  (9)  n  further 
equations, 


(13) 


There  thus  exist  (mn  +  n)  equations  for  the  determination 
of  the  mn  constants  A*  and  the  m  constants  ai}  or  altogether 
(mn  +  m)  constants.  That  is, 


and 
where 


m  =  n 

n 

i*  =  i'  +  2)1'  A*  £~a*> 
i 


Af 


<„•; 


and 


(14) 
(15) 

(16) 
(17) 

(18) 


GENERAL  SYSTEM  OF  CIRCUITS 


171 


Each  of  the  n  sets  of  n  linear  homogeneous  equations  in 
A*  (16)  which  contains  the  same  index  i  gives  by  elimination 
of  At"  the  same  determinant : 


-a^1,  bf-atf,  b^- 


Thus  the  n  values  of  at  are  the  n  roots  of  the  equation  of  nth 
degree  (19),  and  determined  by  solving  this  equation. 

Substituting  these  n  values  of  at  in  the  equations  (16)  gives 
n2  linear  homogeneous  equations  in  A/,  of  which  n  (n  —  1)  are 
independent  equations,  and  these  n  (n  —  1)  independent  equa- 
tions together  with  the  n  equations  (17)  give  the  n2  linear 
equations  required  for  the  determination  of  the  n2  con- 
stants A*. 

The  problem  of  determining  the  equations  of  the  phenomena 
in  starting,  or  in  any  other  way  changing  the  circuit  conditions, 
in  a  general  system  containing  only  resistance  and  inductance, 
with  n  independent  currents  and  such  impressed  e.m.fs.,  ey, 
that  the  equations  of  stationary  condition, 


can  be  solved,  still  depends  upon  the  solution  of  an  equation  of 
nth  degree,  in  the  exponents  at  of  the  exponential  functions 
which  represent  the  transient  term. 

96.  As  an  example  of  the  application  of  this  method  may 
be  considered  the  following  case,  sketched  diagrammatically  in 
Fig.  42: 

An  alternator  of  e.m.f.  E  cos  (6  -  00)  feeds  over  resistance 
rl  the  primary  of  a  transformer  of  mutual  reactance  xm.  The 
secondary  of  this  transformer  feeds  over  resistances  r2  and  rs 
the  primary  of  a  second  transformer  of  mutual  reactance  xmo, 
and  the  secondary  of  this  second  transformer  is  closed  by  resist- 
ance r4.  Across  the  circuit  between  the  two  transformers  and 
the  two  resistances  r2  and  r3,  is  connected  a  continuous-current 


172 


TRANSIENT  PHENOMENA 


e.m.f.,  e0,  as  a  battery,  in  series  with  an  inductive  reactance  x. 
The  transformers  obviously  must  be  such  as  not  to  be  saturated 
magnetically  by  the  component  of  continuous  current  which 
traverses  them,  must  for  instance  be  open  core  transformers. 


Fig.  42.     Alternating-current  circuit  containing  mutual  and  self-inductive 
reactance,  resistance  and  continuous  e.m.f. 


Let  iv  iv  iw  is,  i4  =  currents  in  the  different  circuits;  then,  at 
the  dividing  point  P,  by  equation  (2)  we  have 


hence,  iQ  =  i3  —  i2, 

leaving  four  independent  currents  iv  i2,  i3,  i^. 
This  gives  four  equations  (4) : 


E 


M 


-e- 


=0, 


Xm     ,  _     ~\      X   (   ~~~ 

,dd 
•x\**-      ^1 

"^    \  JQ  JQ    I    ~    U? 


(20) 


and 


(21) 


If  now  i/,  i2',  i',,  t/  are  the  permanent  terms  of  current,  by 
substituting  these  into  (21)  and  subtraction,  the  equations  of 
the  transient  terms  rearranged  are : 


q:    K  = 

1 

2 
3 


GENERAL  SYSTEM  OF  CIRCUITS 
234 


173 


di2 

__  nt     £      I      If    n          I      /v. 

*        7rt         I      '    •*»      1^'*' 


do 

din 


0, 


=  0, 


^'4 

W 


0. 


(22) 


These  equations  integrated  by 


.-a, a 


(23) 


give  for  the  determination  of  the  exponents  c^  the  determinant 

(19): 

r,       -  axm          0  0 


0 

0  ax         r3  —  ax      -  axmo 


0  0 


-ax 


mo 


=  0; 


(24) 


or,  resolved, 


-  a 


-  axrxr4  (ra  +  ra)  +  r1rar8r4  =  0. 

Assuming  now  the  numerical  values, 
rt  =     1 


(25) 


r,  «    1 


xm  =      10 

X™Q  =      10 

«  =  100 
r4  =  10 

equation  (25)  gives 

/  =  a4  +  11  a3  -  0.11  a2  -  0.2  a  +  0.001  =  0. 

The  sixteen  coefficients, 

A?,    i  =  1,  2,  3,  4,    ft  =  1,  2,  3,  4, 

are  now  determined  by  the  16  independent  linear  equations  (12) 
and  (13). 


(27) 


1T4  TRANSIENT  PHENOMENA 

(B)  Circuits  containing  resistance,  self -inductance,  mutual  in- 
ductance and  capacity. 

97.  The  general  method  of  dealing  with  such  a  system  is  the 
same  as  in  (A). 

Kirchhoff's  equation  (1)  is  of  the  form 

i 

dt  =  0.         (28) 

Eliminating  now  all  the  currents  which  can  be  expressed 
in  terms  of  other  currents,  by  means  of  equation  (2),  leaves 
n  independent  currents : 

iK,    K  =  1,  2, . . .  n. 

Substituting  these  currents  iK  in  equations  (28)  gives  n  inde- 
pendent equations  of  the  form 

n  n  7  •  n 

eq  -  X"  &A  -    X"  c«'-ir  -    X"  &ff  /  *«  dt  =  °-     (29) 

i  i  i 

Resolving  these  equations  for  /  iK  dt  gives 

e/  =  i  fi*=  2>  +  I>-  +  2;c^        (so) 


as  the  equations  of  the  potential  differences  at  the  condensers. 
Differentiating  (29)  gives 


where  q  =  1,  2,  .  .  .  n. 

By  the  same  reasoning  as  before,  the  solution  of  these  equa- 
tions (31)  can  be  split  into  two  components,  a  permanent  term, 

(32) 


and  a  transient  term,  which  disappears  for  t  =  oo  ,  and  is  given 
by  the  n  simultaneous  differential  equations  of  second  order, 
thus  : 


GENERAL  SYSTEM  OF  CIRCUITS 
These  equations  are  integrated  by 

m 

IK    ^   ^^l        i  ^       '  * 

1 

Substituting  (34)  in  (33)  gives 


where  q  =  1,  2,  . . .  n, 

*  =  1,  2,  .  . .  n, 

and  i  =  1,  2,  .  .  .  m. 

Reversing  in  these  n  equations  the  order  of  summation, 


175 

(34) 

(35) 
(36) 


=  0,         (37) 


and  this  gives,  as  identity,  the  mn  equations  for  the  determina- 
tion of  the  constants  : 


where 


(38) 


q  =  1,  2,  . .  .  n    and    i  =  1,  2,  . .  .  m. 

In  addition  to  these  mn  equations  (38),  two  sets  of  terminal 
conditions  exist,  depending  respectively  on  the  instantaneous 
current  and  the  instantaneous  condenser  potential  at  the  moment 
of  start. 

The  current  is 


and  the  condenser  potential  of  the  circuit  q  is 


hence,  for  i  =  0, 


« o  _  v  /   i    \V  A  * 

*«  -      1K         '      '       Zu        **»     1 


(39) 


' ;    (40) 


(41) 


176  TRANSIENT  PHENOMENA 

where  K  =  1,  2,  .  .  .  n, 

and  e*  =  eq  -  £.  6.V  •-  £  «  c,<  ^°  ,  (42) 

i  i 

where,  q  =  1,  2  ...  n; 

or,  substituting  (39)  in  (40),  and  then  putting  t  =  0, 


)1'  4"(&.«  -  OA4).  (43) 


1  1 

As  seen,  in  (41)  and  (43),  the  first  term  is  the  instantaneous 
value  of  the  permanent  current  i'K  and  condenser  potential  eq'. 

These  two  sets  of  n  equations  each,  given  by  the  terminal 
conditions  of  the  current,  i'K  =  iK°  (42),  and  condenser  potential, 
eq'  =  eq°  (43),  together  with  the  mn  equations  (38),  give  a  total 
of  (mn  +  2  n)  equations  for  the  determination  of  the  mn  con- 
stants A*  and  the  m  constants  ai}  that  is,  a  total  of  (mn  +  m) 
constants. 

From 

mn  +  2  n  =  mn  -f  m 

it  follows  that 

m  =  2  n.  (44) 

We  have,  then,  2  n  constants,  ai}  giving  the  coefficients  in  the 
exponents  of  the  2  n  exponential  transient  terms,  and  2  n2 
coefficients,  A*,  and  for  their  determination  2n2  equations, 

£«  Af  (g*  -  aj>*  +  afcf)  =  0,  (45) 

i 

n  equations, 

t  =  i«,  (46) 


and  n  equations, 

x^ 

i      i 


GENERAL  SYSTEM  OF  CIRCUITS 


where 


ft*!  ; 

«y  J<=o 


177 

(48) 


or  the  difference  between  the  condenser  potential  required  by 
the  permanent  term  and  the  actual  condenser  potential  at  time 
t  =  0}  where 

q  =  1,  2,  3,  .  .  .  n, 


and 


K  =  1,  2,  3,  .  .  .  n, 
i  =  1,  2,  3,  .  .  .  2  n. 


(49) 


Eliminating  A*  from  the  equations  (45)  gives  for  each  of  the 
2  n  sets  of  n  equations  which  have  the  same  at-  the  determinant : 


g3n  - 


=  0.(50) 


The  2  n  values  of  a,  thus  are  the  roots  of  an  equation  of  2  nth 
order. 

Substituting  these  values  of  at-  in  equations  (45),  (46),  (47), 
leaves  2  n  (n  —  1)  independent  equations  (45)  and  2  n  inde- 
pendent equations  (46)  and  (47),  or  a  total  of  2  n2  linear  equa- 
tions, for  the  determination  of  the  2  n2  constants  Aft  which  now 
can  easily  be  solved. 

The  roots  of  equation  (50)  may  either  be  real  or  may  be  com- 
plex imaginary,  and  in  the  latter  case  each  pair  of  conjugate 
roots  gives  by  elimination  of  the  imaginary  form  an  electric 
oscillation. 

That  is,  the  solution  of  the  problem  of  n  independent  circuits 
leads  to  n  transient  terms,  each  of  which  may  be  either  an 
oscillation  or  a  pair  of  exponential  functions. 

98.  The  preceding  discussion  gives  the  general  method  of  the 
determination  of  the  transient  phenomena  occurring  in  any 
system  or  net  work  of  circuits  containing  resistances,  self-indue- 


178  TRANSIENT  PHENOMENA 

tances  and  mutual  inductances  and  capacities,  and  impressed  and 
counter  e.m.fs.  of  any  frequency  or  wave  shape,  alternating  or  con- 
tinuous. 

It  presupposes,  however, 

(1)  That  the  solution  of  the  system  for  the  permanent  terms 
of  currents  and  e.m.fs.  is  given. 

(2)  That,  if  the  impressed  e.m.fs.   contain  transient  terms 
depending  upon  the   currents  in  the  system,   these  transient 
terms  of  impressed  or  counter  e.m.fs.  are  given  as  linear  functions 
of  the  currents  or  of  their  differential  coefficients,  that  is,  the 
rate  of  change  of  the  currents. 

(3)  That  resistance,  inductance,  and  capacity  are  constant 
quantities,  and  for  instance  magnetic  saturation  does  not  appear. 

The  determination  of  the  transient  terms  requires  the  solution 
of  an  equation  of  2  nth  degree,  which  is  lowered  by  one  degree 
for  every  independent  circuit  which  contains  no  capacity. 

Thus,  for  instance,  a  divided  circuit  having  capacity  in  either 
branch  leads  to  a  quartic  equation.  A  transmission  line  loaded 
with  inductive  or  non-inductive  load,  when  representing  the 
capacity  of  the  line  by  a  condenser  shunted  across  its  middle, 
leads  to  a  cubic  equation. 


CHAPTER  XII. 

MAGNETIC    SATURATION   AND    HYSTERESIS    IN    ALTERNAT- 
ING-CURRENT  CIRCUITS. 

99.  If  an  alternating  e.m.f.  is  impressed  upon  a  circuit  con- 
taining resistance  and  inductance,  the  current  and  thereby  the 
magnetic  flux  produced  by  the  current  immediately  assume 
their  final  or  permanent  values  only  in  case  the  circuit  is  closed 
at  that  point  of  the  e.m:f.  wave  at  which  the  permanent  current 
is  zero.  Closing  the  circuit  at  any  other  point  of  the  e.m.f.  wave 
produces  a  transient  term  of  current  and  of  magnetic  flux.  So 
for  instance,  if  the  circuit  is  closed  when  the  current  i  should 
have  its  negative  maximum  value  -  70,  and  therefore  the 
magnetic  flux  and  the  magnetic  flux  density  also  be  at  their 
negative  maximum  value  -  ^>0  and  -  (B0  —  that  is,  in  an 
inductive  circuit,  near  the  zero  value  of  the  decreasing  e.m.f. 
wave  —  during  the  first  half  wave  of  e.m.f.  the  magnetic  flux, 
which  generates  the  counter  e.m.f.,  should  vary  from  —  4>0  to 
+  <I>0,  or  by  2  4>0;  hence,  starting  with  0,  to  generate  the  same 
counter  e.m.f.,  it  must  rise  to  +  2  <l>0,  that  is,  twice  its  permanent 
value,  and  so  the  current  i  also  rises,  at  constant  inductance  L, 
from  zero  to  twice  its  maximum  permanent  value,  2  70.  Since 
the  e.m.f.  consumed  by  the  current  during  the  variation  from 
0  to  2  70  is  greater  than  during  the  normal  variation  from  —  70 
to  +  70,  less  .e.m.f.  is  to  be  generated  by  the  change  of  magnetic 
flux,  that  is,  the  magnetic  flux  does  not  quite  rise  to  2  4>0,  but 
remains  below  this  value  the  more,  the  higher  the  resistance  of 
the  circuit.  During  the  next  half  wave  the  e.m.f.  has  reversed, 
but  the  current  is  still  mostly  in  the  previous  direction,  and  the 
generated  e.m.f.  thus  must  give  the  resistance  drop,  that  is,  the 
total  variation  of  magnetic  flux  must  be  greater  than  2  4>0, 
the  more,  the  higher  the  resistance.  That  is,  starting  at  a  value 
somewhat  below  2  4>0,  it  decreases  below  zero,  and  reaches  a 
negative  value.  During  the  third  half  wave  the  magnetic  flux, 
starting  not  at  zero  as  in  the  first  half  wave,  but  at  a  negative 

179 


180  TRANSIENT  PHENOMENA 

value,  thus  reaches  a  lower  positive  maximum,  and  thus  grad- 
ually, at  a  rate  depending  upon  the  resistance  of  the  circuit,  the 
waves  of  magnetic  flux  4>,  and  thereby  current  i,  approach  their 
final  permanent  or  symmetrical  cycles. 

100.  In  the  preceding,  the  assumption  has  been  made  that 
the  magnetic  flux,  <£,  or  the  flux  density,  (B,  is  proportional  to 
the  current,  or  in  other  words,  that  the  inductance,  L,  is  con- 
stant. If  the  magnetic  circuit  interlinked  with  the  electric 
circuit  contains  iron,  and  especially  if  it  is  an  iron-clad  or  closed 
magnetic  circuit,  as  that  of  a  transformer,  the  current  is  not 
proportional  to  the  magnetic  flux  or  magnetic  flux  density,  but 
increases  for  high  values  of  flux  density  more  than  proportional, 
that  is,  the  flux  density  in  the  iron  reaches  a  finite  limiting  value. 
In  the  case  illustrated  above,  the  current  corresponding  to 
double  the  normal  maximum  magnetic  flux,  3>0,  or  flux  density, 
®0,  may  be  many  times  greater  than  twice  the  normal  maximum 
current,  70.  For  instance,  if  the  maximum  permanent  current 
is  70  =  4.5  amperes,  the  maximum  permanent  flux  density, 
(B0  =  10,000,  and  the  circuit  closed,  as  above,  at  that  point  of 
the  e.m.f.  wave  where  the  flux  density  should  have  its  negative 
maximum,  —  &0  =  —  10,000,  but  the  actual  flux  density  is  0, 
during  the  first  half  wave  of  e.m.f.,  the  flux  density,  when 
neglecting  the  resistance  of  the  electric  circuit,  should  rise  from 
0  to  2  (B0  =  20,000,  and  at  this  high  value  of  saturation  the 
corresponding  current  maximum  would  be,  by  the  magnetic 
cycle,  Fig.  43,  200  amperes,  that  is,  not  twice  but  44.5  times 
the  normal  value.  With  such  excessive  values  of  current,  the 
e.m.f.  consumed  by  resistance  would  be  in  general  considerable, 
and  the  e.m.f.  consumed  by  inductance,  and  therefore  the 
variation  of  magnetic  flux  density,  considerably  decreased,  that 
is,  the  maximum  magnetic  flux  density  would  not  rise  to  20,000, 
but  remain  considerably  below  this  value.  The  maximum 
current,  however,  would  be  still  very  much  greater  than  twice 
the  normal  maximum.  That  is,  in  an  iron-clad  circuit,  in 
starting,  the  transient  term  of  current  may  rise  to  values  very 
much  higher  than  in  air  magnetic  circuits.  While  in  the  latter 
it  is  limited  to  twice  the  normal  value,  in  the  iron-clad  circuit, 
if  the  magnetic  flux  density  reaches  into  the  range  of  magnetic 
saturation,  very  much  higher  values  of  transient  current  are 
found.  Due  to  the  far  greater  effect  of  the  resistance  with  such 


MAGNETIC  SATURATION  AND  HYSTERESIS  181 

excessive  values  of  current,  the  transient  term  of  current  during 
the  first  half  waves  decreases  at  a  more  rapid  rate ;  due  to  the 
lack  of  proportionality  between  current  and  magnetic  flux 
density,  the  transient  term  does  not  follow  the  exponential  law 
any  more. 

101.  In  an  iron-clad  magnetic  circuit,  the  current  is  not  only 
not  proportional  to  the  magnetic  flux  density,  but  the  same 
magnetic  flux  density  can  be  produced  by  different  currents,  or 
with  the  same  current  the  flux  density  can  have  very  different 
values,  depending  on  the  point  of  the  hysteresis  cycle.  Therefore 
the  magnetic  flux  density  for  zero  current  may  equal  zero,  or,  on 
the  decreasing  branch  of  the  hysteresis  cycle,  Fig.  43,  may  be 
+  7600,  or,  on  the  increasing  branch,  —  7600.  Thus,  when 
closing  the  electric  circuit  energizing  an  iron-clad  magnetic 
circuit,  as  a  transformer,  at  the  moment  of  zero  current,  the 
magnetic  flux  density  may  not  be  zero,  but  may  still  have  a  high 
value,  as  remanent  magnetism.  For  instance,  closing  the 
circuit  at  the  point  of  the  e.m.f.  wave  where  the  permanent 
wave  of  magnetic  flux  density  would  have  its  negative  maximum 
value,  —  OJ0  =  --  10,000,  the  actual  density  at  this  moment  may 
be  <$>r  =  +  7600,  the  remanent  magnetism  of  the  cycle.  During 
the  first  half  wave  of  impressed  e.m.f.  the  variation  of  flux 
density  by  2  (B0,  as  required  to  generate  the  counter  e.m.f.,  when 
neglecting  the  resistance,  would  bring  the  positive  maximum  of 
flux  density  up  to  (Br  +  2  (B0  =  27,600,  requiring  1880  amperes 
maximum  current,  or  420  times  the  normal  current.  Obviously, 
no  such  rise  could  occur,  since  the  resistance  of  the  circuit  would 
consume  a  considerable  part  of  the  e.m.f.,  and  so  lower  the  flux 
density  by  reducing  the  e.m.f.  consumed  by  inductance. 

It  is  obvious,  however,  that  excessive  values  of  transient 
current  may  occur  in  transformers  and  other  iron-clad  magnetic 
circuits. 

102.  When  disconnecting  a  transformer,  its  current  becomes 
zero,  that  is,  the  magnetic  flux  density  is  left  at  the  value  of  the 
remanent  magnetism  ±  (Br,  and  during  the  period  of  rest  more 
or  less  decreases  spontaneously  towards  zero.  Hence,  in  con- 
necting a  transformer  into  circuit  its  flux  density  may  be  any- 
where between  +  ($>r  and  -  (Br.  The  maximum  magnetic  flux 
density  during  the  first  half  cycle  of  impressed  e.m.f.  therefore  is 
produced  if  the  circuit  is  closed  at  the  moment  where  the  per- 


182  TRANSIENT  PHENOMENA 

manent  value  of  the  flux  density  should  be  a  maximum,  ±  (B0, 
and  the  actual  density  in  this  moment  is  the  remanent  magnetism 
in  opposite  direction,  T  (Br,  and  the  maximum  value  of 
density  which  could  occur  then  is  ±  ((Br  +  2  (B0).  If  therefore 
the  maximum  magnetic  flux  density  <B0  in  the  transformer  is 
such  that  <Br  +  2  (B0  is  still  below  saturation,  the  transient  term 
of  current  cannot  reach  abnormal  values.  At  &  =  16,000,  the 
flux  density  is  about  at  the  bend  of  the  saturation  curve,  and 
the  current  still  moderate.  Estimating  (Br  =  0.75  (B0  as  approx- 
imate value,  (Br  +  2  (B0  =  16,000  thus  gives  ffi  =  5800,  or 
37,500  lines  of  magnetic  flux  per  square  inch. 

In  125-cycle  transformers,  (B0  is  below  5800  or  not  much  above, 
for  reasons  of  heating,  and  this  phenomenon  of  excessive  tran- 
sient currents  in  starting  thus  does  not  appear.  At  60  cycles, 
(B0  is  usually  above  this  value,  and  under  unfavorable  conditions 
considerable  transient  current  may  be  observed.  However, 
for  (Br  =  0,  the  limit  is  (B0  =  8000,  or  51,600  lines  per  square 
inch;  and  since  in  60-cycle  transformers  the  flux  density  rarely 
exceeds  this  value  to  a  great  extent,  and  in  starting  the  remanent 
magnetism  is  rarely  very  high,  this  phenomenon  of  an  excessive 
transient  current  is  not  very  marked.  At  25  cycles,  however, 
higher  densities  are  used  and  -the  transient  starting  current 
may  then  reach  formidable  values. 

103.  Since  the  relation  between  the  current,  i,  and  the  mag- 
netic flux  density,  (B,  is  empirically  given  by  the  magnetic  cycle 
of  the  material,  and  cannot  be  expressed  with  sufficient  accuracy 
by  a  mathematical  equation,  the  problem  of  determining  the 
transient  starting  current  of  a  transformer  cannot  be  solved  in 
general,  but  must  be  investigated  in  the  individual  case  by 
constructing  the  curves  of  current  and  magnetic  flux  density. 

Let  the  normal  magnetic  cycle  .of  a  transformer  be  represented 
by  the  dotted  curve  in  Figs.  43  and  44;  the  characteristic  points 
are:  the  maximum  values,  ±  OS0  =  ±  10,000;  the  remanent 
values,  ±  (Br  =  ±  7600,  and  the  maximum  exciting  current, 
im  =  ±  4.5  amp. 

At  very  high  values  of  flux  density  an  appreciable  part  of  the 
total  magnetic  flux  $  may  be  carried  through  space,  outside  of 
the  iron,  depending  on  the  construction  of  the  transformer. 
The  most  convenient  way  of  dealing  with  such  a  case  is  to 
resolve  the  magnetic  flux  density,  (B,  in  the  iron  into  the  "  metallic 


MAGNETIC  SATURATION  AND  HYSTERESIS 


183 


600—  WIOCO-  -1400-  -1800-  -2200 


Fig.  43.     Magnetic  cycle  of  a  transformer  starting  with  low  stray  field. 


Fig.  44.     Magnetic  cycle  of  a  transformer  starting  with  high  stray  field. 


184 


TRANSIENT  PHENOMENA 


flux  density,"  ($>'  =  &  -  X,  which  reaches  a  finite  limiting 
value,  and  the  density  in  space,  oe.  The  total  magnetic  flux 
then  consists  of  the  flux  carried  by  the  molecules  of  the  iron, 
$>'  =  A'(B',  where  A'  is  the  section  of  the  iron  circuit,  and  the 
space  flux,  $"  =  A"3C,  where  A"  is  the  total  section  interlinked 
with  the  electric  circuit,  including  iron  as  well  as  other  space. 


If  then  A"  =  &A/,  that  is,  the  total  space  inside  of  the  coil  is 
k  times  the  space  filled  by  the  iron,  we  have 

$  -  A'  (&'  +  te), 

or  the  total  magnetic   flux  even  in  a  case  where  considerable 
stray  field  exists,  that  is,  magnetic  flux  can  pass  also  outside  of 


^m 


-55 
500  — 

-45 
400- 


-25 

200- 
-15 

100- 
-5 
-0 
—5 


100   200   300   400   500   600   700   800   900  1000 
Degrees 


Fig.  45.     Starting  current  of  a  transformer.     Low  stray  field. 

the  iron,  can  be  calculated  by  considering  only  the  iron  section 
as  carrying  magnetic  flux,  but  using  as  curve  of  magnetic  flux 
density  not  the  usual  curve, 

(B  =  &'  +  3C, 
but  a  curve  derived  therefrom, 

(B  =  &'  +  AflC, 

where  k  =  ratio  of  total  section  to  iron  section. 

This,  for  instance,  is  the  usual  method  of  calculating  the 
m.m.f.  consumed  in  the  armature  teeth  of  commutating  machines 
at  very  high  saturations. 


MAGNETIC  SATURATION  AND  HYSTERESIS 


185 


In  investigating  the  transient  transformer  •  starting  current, 
the  magnetic  density  curve  thus  is  corrected  for  the  stray  field. 

Figs.  43  and  45  correspond  to  k  =  3,  or  a  total  effective  air 
section  equal  to  three  times  the  iron  section,  that  is,  (fc  =  (&'  + 

3oe. 

Figs.  44  and  46  correspond  to  k  =  25,  or  a  section  of  stray 
field  equal  to  25  times  the  iron  section,  that  is,  &  =  (&'  +  25  JC. 


22 


,16 


12 


-2 


L 


-CO 


45 


250-25 
200-20 


150-15 
100-10 


50-5 


-5 


100  200  300  400  500  600  700 

Degrees 

Fig.  46.     Starting  current  of  a  transformer.     High  stray  field. 

104.  At  very  high  values  of  current  the  resistance  consumes 
a  considerable  voltage,  and  thus  reduces  the  e.m.f.  generated 
by  the  magnetic  flux,  and  thereby  the  maximum  magnetic  flux 
and  transient  current.  The  resistance,  which  comes  into  con- 
sideration here,  is  the  total  resistance  of  the  transformer  primary 
circuit  plus  leads  and  supply  lines,  back  to  the  point  where  the 
voltage  is  kept  constant,  as  generator,  busbars,  or  supply  main. 

Assuming  then  at  full  load  of  im  =  50  amperes  effective  in  the 
transformer,  a  resistance  drop  of  8  per  cent,  or  the  voltage  con- 
sumed by  the  resistance,  as  er  =  0.08  of  the  impressed  e.m.f. 

Let  now  the  remanent  magnetic  flux  density  be  &r  =  +  7600, 
and  the  circuit  be  closed  at  the  moment  0  =  0,  where  the  flux 


186  TRANSIENT  PHENOMENA 

density  should  be  <&  =  —  <B0  =  —  10,000;  then  the  impressed 
e.m.f.  is  given  by 

e  =  -  E  sin  0  =  E  ^  (cos  0).  (1) 

au 

It  is,  however, 

(KB 

e  =  A  —  +  Ci,  (2) 

where  A  and  C  are  constants;  that  is,  the  impressed  e.m.f.,  e,  is 
consumed  by  the  self-inductance,  or  the  e.m.f.  generated  by  the 

changing  magnetic  density,  which  is  proportional  to  —  ,  and  by 

the  voltage  consumed  by  the  resistance,  which  is  proportional 
to  the  current  i. 
Combining  (1)  and  (2)  gives 


p 

However,  at  full  load,  we  have  -^—  =  effective  impressed 

v2 

e.m.f.  and  im  =  50  amperes  =  effective  current;  hence 
Cim  =  50C  =  e.m.f.  consumed  by  resistance, 
and  since  this  equals  er  =  0.08  of  impressed  e.m.f., 


V2 
0.08 


or 


E       imV2        50 
From  (3)  follows 


:  =  0.00113.  (4) 


d<S>=^dcosd -^dO  (5) 

A  A 


and 

It  IT 

-  |  f*d  cos  0  - '%  f*idd; 

A    J  n  4»^n 


A 


MAGNETIC  SATURATION  AND  HYSTERESIS  187 

hence,  for  i  =  0,  or  negligible  resistance  drop,  that  is,  permanent 
condition, 

».-'|'-io,poa  (6) 

Multiplying  (4)  and  (6)  gives 

?  =  ^7=  =  H-3,  (7) 


and  substituting  (6)  and  (7)  in  (5)  gives 
d&  =  (Bdcos0  -  i 


=  10,000  d  cos  0  -  11.3  idB.  (8) 

Changing  now  from  differential  to  difference,  that  is,  replacing, 
as  approximation,  d  by  A,  gives 

eJB 

A(B  =  (BnA  cos  d  -i  7^—^-^d 

lmV2 

=  10,000  A  cos  0  -  11.3  t'A0.  (9) 

Assuming  now 

A0  =  10°  =  0.175  (10) 

gives  for  the  increment  of  magnetic  flux  density  during  10° 
change  of  angle  the  value 

ACS  -  10,000  A  cos  0  -  2  i  (11) 

and  &  =  <B'  +  A(B 

=  &'  +  10,000  A  cos  d  -  2  i.  (12) 

From  equation  (12)  the  instantaneous  values  of  magnetic 
flux  density  ®,  and  therefrom,  by  the  magnetic  cycles,  Figs.  43 
and  44,  respectively,  the  values  of  current  i  are  calculated,  by 
starting,  for  0  =  0,  with  the  remanent  density  CB'  =  (Br  =  7600, 
adding  thereto  the  change  of  cosine,  10,000  A  cos  0,  which  gives 
a  value  (Bt  =  &'  +  10,000  A  cos  0,  taking  the  corresponding 
value  of  i  from  the  hysteresis  cycle,  Figs.  43  and  44,  subtracting 
2  i  from  (Bt,  and  then  correcting  i  for  the  value  corresponding  to 
<B  =  (Bj  -  2  i. 

The  quantity  2  i  is  appreciable  only  during  the  range  of  the 
curve  where  i  is  very  large. 


188 


TRANSIENT  PHENOMENA 


105.  The  following  table  is  given  to  illustrate  the  beginning 
of  the  calculation  of  the  curve  for  low  stray  field. 

STARTING    CURRENT    OF    A   TRANSFORMER. 


0° 

cos  6 

A(&  = 

10  A  cos  0 

(&- 

(B  +  A(B, 

tm 

PI- 

2  1  X  10~3 

(B 

o 

+  1  00 

7  6 

o 

10 

0  98 

+  0.2 

7.8 

1  0 

20 
30 

0.94 
0.87 

0.4 
0.7 

8.2 
8.9 

1.9 
2  9 



40 

0.77 

1.0 

9.9 

3  8 

50 

0  64 

1.3 

11  2 

5  2 

60 

0.50 

1.4 

12.6 

7.3 

70 
80 
90 
100 
110 
120 
130 
140 
150 
160 
170 
180 
190 
200 
210 
220 
230 
240 
250 
260 
270 

0.34 
+  0.17 
0 
-0.17 
0.34 
0.50 
0.64 
0.77 
0.87 
0.94 
0.98 
1.00 
0.98 
0.94 
0.87 
0.77 
0.64 
0.50 
0.34 
-0.17 
0 

1.6 

.7 
.7 
.7 
.7 
.6 
.4 
.3 
.0 
0.7 
0.4 
+  0.2 
-0.2 
0.4 
0.7 
1.0 
.3 
.4 
.6 
.7 
.7 

14.2 
15.9 
17.6 
19.2 
20.6 
21.75 
22.6 
23.0 
23.0 
22.7 
22.2 
21.7 
21.0 
20.2 
19.15   . 
17.9 
16.45 
14.85 
13.2 
11.5 
9.8 

12.0 
27 
70 
138 
220 
270 
450 
510 
510 
440 
350 
250 
200 
180 
130 
76 
40 
14 
7 
3.2 
-1.0 

0.1 
0.3 
0.45 
0.55 
0.9 
1.0 
1.0 
0.9 
0.7 
0.5 
0.4 
0.35 
0.25 
0.15 
0.2 
0.05 

"i-r.'s" 

18.9 
20.15 
21.2 
21.7 
22.0 
22.0 
21.8 
21.5 
21.2 
20.6 
19.85 
18.9 
17.75 
16.25 
14.8 

280 

+  0.17 

.7 

8.1 

-  .4 

290 

0.34 

.7 

6.4 

-1.3 

300 

0.50 

.6 

4.8 

—  1  9 

310 

0.64 

.4 

3.4 

-2  2 

320 
330 

0.77 
0.87 

.3 
1.0 

2.1 
1.1 

-2.5 
-2  6 



340 

0.94 

0.7 

.4 

-2.75 

350 

0.98 

0.4 

0 

—  2  8 

360 

+  1.00 

-0.2 

-  .2 

—  2  8 

370 

0.98 

+  0.2 

0 

-1  3 

380 
390 
400 

0.94 
0.87 
0.77 

0.4 
0.7 
.0 

+  0.4 
1.1 
2  1 

-0.5 
+  0.3 
1  0 



410 
420 

0.64 
0.50 

.3 
.4 

3.5 
4.9 

1.7 
2  2 

430 
440 
450 

0.34 
+  0.17 
0 

.6 

.7 
.7 

6.5 

8.2 
9.9 

2.7 
3.3 
4  3 

460 
470 
480 

-0.17 
-0.34 
-0.50 

.7 
.7 
.6 

11.6 
13.3 
14.9 

6.2 
9.5 
16.5 

MAGNETIC  SATURATION  AND  HYSTERESIS 


189 


The  first  column  gives  angle  0, 

The  second  column  gives  cos  d, 

The  third  column  gives  A(Bt  =  10  A  cos  0,  in  kilolines  per 
sq.  cm., 

The  fourth  column  gives  0^  =  (B  +  Afl^, 

The  fifth  column  gives  i, 

The  sixth  column  gives  D1  =  2  i  X'lCT3,  and 

The  seventh  column  gives  (B  =  (Bt  —  Dv 

i  in  the  fifth  column  being  chosen,  by  trial,  so  as  to  corre- 
spond, on  the  hysteresis  cycles,  not  to  <&v  but  to  (B  =  (Bt  —  Dr 

These  values  are  recorded  as  magnetic  cycles  on  Figs.  43  and 
44,  and  as  waves  of  flux  density,  current,  etc.,  in  Figs.  45  and  46. 

The  maximum  values  of  successive  half  waves  are : 


A.  Low  Stray  Field. 
fc=  3 

B.  High  Stray  Field. 
fc=  25 

9 

(B 

im 

e 

(B 

*m 

0 
145° 
360° 
530° 
720° 
900° 
1080° 
1260° 
1440° 
1620° 

7.6 
22.0 
-    .2 
18.6 
-2.2 
18.0 
-2.8 
17.4 
-3.1 
16.9 

0 
510 
-2.8 
120 
-2.9 
92 
—  30 

0 

160° 
360° 
530° 
720° 

7.6 
24.6 
+  2.0 
20.7 
-    .4 

0 
230 
-2.4 
117 
-2.7 

66 
-3.0 
50 

Permanent 

±10.0 

±4.5 

±10.0 

±4.5 

As  seen,  the  maximum  value  of  current  during  the  first  cycle, 
510,  is  more  than  one  hundred  times  the  final  value  4.5,  and  more 
than  7  times  the  maximum  value  of  the  full-load  current,  50\/2 
=  70.7  amperes,  and  the  transient  current  falls  below  full-load 
current  only  in  the  fourth  cycle.  That  is,  the  excessive  value  of 
transient  current  in  an  ironclad  circuit  lasts  for  a  considerable 
number  of  cycles. 

In  the  presence  of  iron  in  the  magnetic  field  of  electric  circuits, 
transient  terms  of  current  may  thus  occur  which  are  very  large 
compared  with  the  transient  terms  in  ironless  reactors,  which  do 
not  follow  the  exponential  curve,  can  usually  not  be  calculated 


190 


TRANSIENT   PHENOMENA 


by  general  equations,  but  require  numerical  investigation  by 
the  use  of  the  magnetic  cycles  of  the  iron. 

These  transient  terms  lead  to  excessive  current  values  only  if 
the  normal  magnetic  flux  density  exceeds  half  the  saturation 
value  of  the  iron,  and  so  are  most  noticeable  in  25-cycle  circuits. 


A  !\  l\ 


YYVV 


Fig.  47.     Starting  current  of  a  25-cycle  transformer. 

As  illustration  is  shown,  in  Fig.  47,  an  oscillogram  of  the 
starting  current  of  a  25-cycle  transformer  having  a  resistance 
in  the  supply  circuit  somewhat  smaller  than  in  the  above 
instance,  thus  causing  a  still  longer  duration  of  the  transient 
term  of  excessive  current. 

These  starting  transients  of  the  ironclad  inductance  at  high 
density  are  unsymmetrical  waves,  that  is>  successive  half  waves 
have  different  shapes,  and  when  resolved  into  a  trigonometric 
series,  would  give  even  harmonics  as  well  as  the  odd  harmonics. 

Thus  the  first  wave  of  Fig.  45  can,  when  neglecting  the  tran- 
sient factor,  be  represented  by  the  series: 

i  =   +  108.3  -  183.8  cos  (0  +  28.0°) 

+  112.4  cos  2  (0  +  29.8°)  -  53.1  cos  3  (6  +  33.3°) 
+  27.2  cos  4  (0  +  39.1°)  -  18.4  cos  5  (0  +  38.1°) 
+  13.6  cos  6  (0  +  33.4°)  -  8.1  cos  7  (0  +  32.7°) 
or,  substituting:  6  =  ft  +  150°,  gives: 

e  =  E  sin  (ft  +  150°) 

i  =  108.3  +  183.8  cos  (/?  -  2.0°)  +  112.4  cos  2  (p  -  0.2°) 
+  53.1  cos  3  0?  +  3.3°)  +  27.2  cos  4  (/?  +  9.1°) 
+  18.4  cos  5  (/?  +  8.1°)  +  13.6  cos  6  (p  +  3.4°) 
+  8.1  cos  7  (/?  +  2.7°). 


CHAPTER  XIII. 

TRANSIENT   TERM    OF   THE    ROTATING   FIELD. 

106.  The  resultant  of   np  equal    m.m.fs.   equally  displaced 
from  each  other  in  space  angle  and  in  time-phase  is  constant  in 
intensity,  and  revolves  at  constant  synchronous  velocity.     When 
acting  upon  a  magnetic  circuit  of  constant  reluctance  in  all 
directions,   such   a  polyphase  system   of   m.m.fs.   produces   a 
revolving   magnetic    flux,  or   a  rotating  field.     ("  Theory  and 
Calculation  of  Alternating  Current   Phenomena,"  4th  edition, 
Chapter  XXXIII,  paragraph  368.)     That  is,  if  np  equal  mag- 
netizing coils  are  arranged  under  equal  space  angles  of  - 

np 

electrical  degrees,  and  connected  to  a  symmetrical  np  phase 
system,  that  is,  to  np  equal  e.m.fs.  displaced  in  time-phase  by 

360 

-  degrees,  the  resultant  m.m.f.  of  these  np  coils  is  a  constant 

np 

and  uniformly  revolving  m.m.f.,  of  intensity  SF0  =    —  &,  where  $ 

Zi 

is  the  maximum  value  (hence  — —  the  effective  value)  of  the 

\  V2  I 

m.m.f.  of  each  coil. 

In  starting,  that  is,  when  connecting  such  a  system  of  mag- 
netizing coils  to  a  polyphase  system  of  e.m.fs.,  a  transient  term 
appears,  as  the  resultant  magnetic  flux  first  has  to  rise  to  its 
constant  value.  This  transient  term  of  the  rotating  field  is  the 
resultant  of  the  transient  terms  of  the  currents  and  therefore 
the  m.m.fs.  of  the  individual  coils. 

107.  If,  then,  $  =  nl  =  maximum  value  of  m.m.f.  of  each 
coil,  where  n  =  number  of  turns,  and  /  =  maximum  value  of 
current,  and  r  =  space-phase  angle  of  the  coil,  the  instantaneous 
value  of  the  m.m.f.  of  the  coil,  under  permanent  conditions,  is 

f-frCOS^-T),  (1) 

191 


192  TRANSIENT  PHENOMENA 

and  if  the  time  6  is  counted  from  the  moment  of  closing  the 
circuit,  the  transient  term  is,  by  Chapter  IV, 

/"  =    -  gr£~  x  0  cos  T}  (2) 

where  Z  =  r  —  jx. 

The  complete  value  of  m.m.f.  of  one  coil  is 

/i  =  /'  +  /"  =  &  {cos(0  -  r)  -r-'cosr}.          (3) 
In  an  np-phase  system,  successive  e.m.fs.  and  therefore  currents 

are  displaced  from  each  other  by  —  of  a  period,  or  an  angle  —  , 

np  np 

and  the  m.m.f.  of  coil,  i,  thus  is 

(         /  2n  \         -r~6         I        2?r   \) 

ff  =  $  <  cos  (0  —  T  —  •  —  i    —  e    x    cos  (  r  H  --  i]>.    (4) 
(         \  n      I  \        n     n 


p 


/         2  TT   \ 

2ji  cos   T  H  --  i)  =  0; 
\        nv    I 


The  resultant  of  np  such  m.m.fs.  acting  together  in  the  same 
direction  would  be 


(5) 


that  is,  the  sum  of  the  instantaneous  values  of  the  permanent 
terms  as  well  as  the  transient  terms  of  all  the  phases  of  a  sym- 
metrical polyphase  system  equals  zero. 

In  the  polyphase  field,  however,  these  m.m.fs.  (4)  do  not  act 
in  the  same  direction,  but  in  directions  displaced  from  each 

other  by  a  space  angle  —  equal  to  the  time  angle  of  their  phase 
np 

displacement. 

108.  The  component  of  the  m.m.f.,  fit  acting  in  the  direction 
(00  -  T),  thus  is 

27T      .> 


//  =  ft  cos  (»„  -  T  -  ~  i),  (6) 

\  nn     i 


TRANSIENT   TERM  OF   THE  ROTATING  FIELD          193 

and  the  sum  of  the  components  of  all  the  np  m.m.fs.,  in  the 
direction  (00  -  r),  that  is,  the  component  of  the  resultant  m.m.f. 
of  the  polyphase  field,  in  the  direction  (00  -  T),  is 

np 

f  =  X<  fi 

I 


Transformed,  this  gives 

$  (   np          /  4  TT    \       "* 

/  =    ;     !  5/  cos  (0  +  00  -  2  r  -       -  i)  +  V»   c 

^       I          i^  \  "^n          / 

-Htf   Hp  I 

cos  0ft  -  s    x    Vt  cos  ( #n  —  2  T  —  — 


r.    "P  47r 


and  as  the  sums  containing  -  -  i  equal  zero,  we  have 

np 


os(0-00)  -cos00,  (8) 

and  for  6  =  oo  f  that  is  as  permanent  term,  this  gives 

/0=^SFcos(0-00);  (9) 

ft 

hence,  a  maximum,  and  equal  to  -~  &,  that  is,  constant,  for 

00  =  0,  that  is,  uniform  synchronous  rotation.  That  is,  the 
resultant  of  a  polyphase  system  of  m.m.fs.,  in  permanent  con- 
dition, rotates  at  constant  intensity  and  constant  synchronous 
velocity. 

Before  permanent  condition  is  reached,  however,  the  resultant 
m.m.f.  in  the  direction  #0  =  6,  that  is,  in  the  direction  of  the 
synchronously  rotating  vector,  in  which  in  permanent  condition 


194 


TRANSIENT  PHENOMENA 


the  m.m.f .  is  maximum  and  constant,  is  given  during  the  transient 
period,  from  equation  (8),  by 


(10) 


that  is,  it  is  not  constant  but  periodically  varying.  • 

As  example  is  shown,  in  Fig.  48,  the  resultant  m.m.f.  /0  in  the 
direction  of  the  synchronously  revolving  vector,  00  =  6,  for  the 


1600 

fnoo 

6  800 

a 

400 

r 

\ 

I 

08'( 

\ 

s~ 

N, 

fn 

=  ' 

mBi 

_( 

-0. 

cos  6 

[ 

j 

\ 

^ 

X, 

1 

\ 

1 

\ 

1 

S 

^ 

^  — 

^v 

s 

I 

\ 

1 

\ 

/ 

s 

s 

^ 

/ 

^.^ 

1 

^ 

s 

1 

V 

y 

/ 

-V 

0 

/ 

27T 

37T 

4 

7T 

5 

7T 

c 

7T 

7 

7T 

87T 

90  180  270  360  450  540  630  720  810  900  990  1060 1170 1260 1350 1440 

Fig.  48.     Transient  term  of  polyphase  magnetomotive  force. 

constants    np  =3,    or   a   three-phase   system;   SF  =  667,    and 
Z  =  r  -  jx  =  0.32  -  4  j ;  hence, 


/0  -  1000  (1  -  £ 


—  1.080 


cos  d), 


with  6  as  abscissas,  showing  the  gradual  oscillatory  approach  to 
constancy. 

109.  The  direction,  6Q  =  0,  is,  however,  not  the  direction  in 
which  the  resultant  m.m.f.  in  equation  (8)  is  a  maximum,  but 
the  maximum  is  given  by 


df 


=  0, 


this  gives 
hence, 


sin  (0  -  60)  +  s    x    sin  00  =  0, 


cot  0n  = 


COS  0   —  £ 


sin  0 


(11) 

(12) 

(13) 


that  is,  the  resultant  maximum  m.m.f.  of  the  polyphase  system 
does  not  revolve  synchronously,  in  the  starting  condition,  but 
revolves  with  a  varying  velocity,  alternately  running  ahead  and 


TRANSIENT  TERM  OF   THE  ROTATING  FIELD  195 

dropping  behind  the  position  of  uniform  synchronous  rotation, 
by  equation  (13),  and  only  for  0  =  oo,  equation  (12)  becomes 
cot  #0  =  cot  6,  or  #0  =  6,  that  is,  uniform  synchronous  rotation. 
The  speed  of  rotation  of  the  maximum  m.m.f.  is  given  from 
equation  (12)  by  differentiation  as 

dQ 

^_0=       £ 
dd  '        dQ' 


where  Q  =  sin  (6  —  00)  +  £    x  sin  00; 

_  -  a 

cos  {0  -  00)  -  -£    x   sin  00 

hence,  S  =  -^—        ~,  (14) 

cos  (0  -  00)  -  e~xfl  cos  00 

or  approximately, 


1    -  £      x      COS  6 


(15) 


For  0  =  oo,  equation  (14)  becomes  S  =  1,  or  uniform  syn- 
chronous rotation,  but  during  the  starting  period  the  speed 
alternates  between  below  and  above  synchronism. 

From  (13)  follows 


-- 

COS  0  —  £      x 


and 


sin  d 
sin  0    = 


(R 
where 


(16) 


\ 

(R  =  V     os    -  e     z        +  sn      =          -£    '    cos         e 

(17) 


196  TRANSIENT  PHENOMENA 

110.  The  maximum  value  of  the  resultant  m.m.f.,  at  time- 
phase  6,  and  thus  of  direction  6Q  as  given  by  equation  (13)  or 
(16),  (17),  is  derived  by  substituting  (16),  (17)  into  (8),  as: 


-2£    *os0  +  e     *,  (18) 

hence  is  not  constant,  but  pulsates  periodically,  with  gradually 

decreasing  amplitude  of  pulsation,  around  the  mean  value  —  $. 

2i 

For  6  =  0,  or  at  the  moment  of  start,  it  is,  by  (13), 

-r-e 
cos  6  -  £  *         0 


'  sinfl  Q 

hence,  differentiating  numerator  and  denominator, 


r  -- 
—sin  6  +  -s   x 


X 

r 

cos  0 

x 

tan00'=?; 

cot  007  = 
and 


that  is,  the  position  of  maximum  resultant  m.m.f.  starts  from 
angle  00'  ahead  of  the  permanent  position,  where  00'  is  the  time- 
phase  angle  of  the  electric  magnetizing  circuit.  The  initial 
value  of  the  resultant  m.m.f.,  for  0  =  0,  is  fm  =  0,  that  is,  the 
revolving  m.m.f.  starts  from  zero. 
Substituting  (16)  in  (15)  gives  the  speed  as  function  of  time 


(cos  6  --  sin  0) 


(19) 


1  +  s~*—2e~*    cos# 
for  0  =  0  this  gives  the  starting  speed  of  the  rotating  field 

SQ  =  -  ,  or,  indefinite; 


TRANSIENT   TERM  OF   THE  ROTATING  FIELD 


197 


hence,  after  differentiating  numerator  and  denominator  twice, 
this  value  becomes  definite. 

So  =  ^;  (20) 

that  is,  the  rotating  field  starts  at  half  speed. 

As  illustration  are  shown,  in  Fig.  49,  the  maximum  value  of 
the  resultant  polyphase  m.m.f.,  fm,  and  its  displacement  in 


+40 

'     ° 
-40 


,400 


Inten'sit^/ 


=  1000 


/\ 


\f 


position  |(0 


ili 

+  8il 


mil 


2  2  2  22 

Fig.  49.     Start  of  rotating  field. 


position  from  that  of  uniform  synchronous  rotation,  00—  6,  for 
the  same  constants  as  before,  namely:  np  =  3;  *F  =  667,  and 
Z  =  r  -  jx  =  0.32  -  4  /;  hence, 


fm   =   1000  Vl    -  2  £-C        '     COS  0  +  £-C  •», 

with  the  time-phase  angle  ^  as  abscissas,  for  the  first  three  cycles. 
111.  As  seen,  the  resultant  maximum  m.m.f.  of  the  poly- 
phase system,  under  the  assumed  condition,  starting  at  zero 
in  the  moment  of  closing  the  three-phase  circuit,  rises  rapidly 
-  within  60  time-degrees  —  to  its  normal  value,  overreaches 
and  exceeds  it  by  78  per  cent,  then  drops  down  again  below 
normal,  by  60  per  cent,  rises  47  per  cent  above  normal,  drops 
37  per  cent  below  normal,  rises  28  per  cent  above  normal,  and 
thus  by  a  series  of  oscillations  approaches  the  normal  value. 
The  maximum  value  of  the  resultant  m.m.f.  starts  in  position 


198  TRANSIENT  PHENOMENA 

85  time-degrees  ahead,  in  the  direction  of  rotation,  but  has  in 
half  a  period  dropped  back  to  the  normal  position,  that  is,  the 
position  of  uniform  synchronous  rotation,  then  drops  still  fur- 
ther back  to  the  maximum  of  40  deg.,  runs  ahead  to  34  deg., 
drops  23  deg.  behind,  etc. 

It  is  interesting  to  note  that  the  transient  term  of  the  rotat- 
ing field,  as  given  by  equations  (10),  (13),  (18),  does  not  contain 
the  phase  angle,  that  is,  does  not  depend  upon  the  point  of  the 
wave,  0  =  r,  at  which  the  circuit  is  closed,  while  in  all  preced- 
ing investigations  the  transient  term  depended  upon  the  point 
of  the  wave  at  which  the  circuit  was  closed,  and  that  this  tran- 
sient term  is  oscillatory.  In  the  preceding  chapter,  in  circuits 
containing  only  resistance  and  inductance,  the  transient  term 
has  always  been  gradual  or  logarithmic,  and  oscillatory  phenom- 
ena occurred  only  in  the  presence  of  capacity  in  addition  to  in- 
ductance. In  the  rotating  field,  or  the  polyphase  m.m.f.,  we 
thus  have  a  case  where  an  oscillatory  transient  term  occurs  in 
a  circuit  containing  only  resistance  and  inductance  but  not 
capacity,  and  where  this  transient  term  is  independent  of  the 
point  of  the  wave  at  which  the  circuits  were  closed,  that  is,  is 
always  the  same,  regardless  of  the  moment  of  start  of  the  phe- 
nomenon. 

The  transient  term  of  the  polyphase  m.m.f.  thus  is  independ- 
ent of  the  moment  of  start,  and  oscillatory  in  character,  with 
an  amplitude  of  oscillation  depending  only  on  the  reactance 

factor,  — ,  of  the  circuit. 


CHAPTER  XIV. 

SHORT-CIRCUIT  CURRENTS   OF   ALTERNATORS. 

112.  The  short-circuit  current  of  an  alternator  is  limited  by 
armature  reaction  and  armature  self-inductance;  that  is,  the 
current  in  the  armature  represents  a  m.m.f.  which  with  lagging 
current,  as  at  short  circuit,  is  demagnetizing  or  opposing  the 
impressed  m.m.f.  of  field  excitation,  and  by  combining  therewith 
to  a  resultant  m.m.f.  reduces  the  magnetic  flux  from  that  corre- 
sponding to  the  field  excitation  to  that  corresponding  to  the 
resultant  of  field  excitation  and  armature  reaction,  and  thus 
reduces  the  generated  e.m.f.  from  the  nominal  generated  e.m.f., 
eOJ  to  the  virtual  generated  e.m.f.,  er  The  armature  current 
also  produces  a  local  magnetic  flux  in  the  armature  iron  and  pole- 
faces  which  does  not  interlink  with  the  field  coils,  but  is  a  true 
self-inductive  flux,  and  therefore  is  represented  by  a  reactance  xr 
Combined  with  the  effective  resistance,  rv  of  the  armature 
winding,  this  gives  the  self-inductive  impedance  Zl  =  rl  — 


or  zt  =  Vr*  +  x*.  Vectorially  subtracted  from  the  virtual 
generated  e.m.f.,  ev  the  voltage  consumed  by  the  armature 
current  in  the  self-inductive  impedance  Zl  then  gives  the  ter- 
minal voltage,  e. 

At  short  circuit,  the  virtual  generated  e.m.f.,  ev  is  consumed 
by  the  armature  self-inductive  impedance,  zr  As  the  effective 
armature  resistance,  rv  is  very  small  compared  with  its  self- 
inductive  reactance,  xv  it  can  be  neglected  compared  thereto, 
and  the  short-circuit  current  of  the  alternator,  in  permanent 
condition,  thus  is 


As  shown  in  Chapter  XXII,  "Theory  and  Calculation  of 
Alternating  Current  Phenomena,"  the  armature  reaction  can  be 
represented  by  an  equivalent,  or  effective  reactance,  z2,  and  the 
self-inductive  reactance,  xv  and  the  effective  reactance  of 

199 


200  TRANSIENT  PHENOMENA 

armature  reaction,  x2J  combine  to  form  the  synchronous  react- 
ance, XQ  =  xl  +  x2,  and  the  short-circuit  current  of  the  alterna- 
tor, in  permanent  condition,  therefore  can  be  expressed  by 


where  e0  =  nominal  generated  e.m.f. 

113.  The  effective  reactance  of  armature  reaction,  xv  differs, 
however,  essentially  from  the  true  self-inductive  reactance,  xv 
in  that  xl  is  instantaneous  in  its  action,  while  the  effective 
reactance  of  armature  reaction,  xv  requires  an  appreciable  time 
to  develop:  x2  represents  the  change  of  the  magnetic  field  flux 
produced  by  the  armature  m.m.f.  The  field  flux,  however,  can- 
not change  instantaneously,  as  it  interlinks  with  the  field  exciting 
coil,  and  any  change  of  the  field  flux  generates  an  e.m.f.  in  the 
field  coils,  changing  the  field  current  so  as  to  retard  the  change 
of  the  field  flux.  Hence,  at  the  first  moment  after  a  change  of 
armature  current,  the  current  change  meets  only  the  reactance, 
xv  but  not  the  reactance  x2.  Thus,  when  suddenly  short-cir- 
cuiting an  alternator  from  open  circuit,  in  the  moment  before 
the  short  circuit,  the  field  flux  is  that  corresponding  to  the 
impressed  m.m.f.  of  field  excitation  and  the  voltage  in  the  arma- 
ture, i.e.,  the  nominal  generated  e.m.f.,  e0  (corrected  for  mag- 
netic saturation).  At  the  moment  of  short  circuit,  a  counter 
m.m.f.,  that  of  the  armature  reaction  of  the  short-circuit 
current,  is  opposed  to  the  impressed  m.m.f.  of  the  field  excitation, 
and  the  magnetic  flux,  therefore,  begins  to  decrease  at  such  a 
rate  that  the  e.m.f.  generated  in  the  field  coils  by  the  decrease 
of  field  flux  increases  the  field  current  and  therewith  the  m.m.f. 
so  that  when  combined  with  the  armature  reaction  it  gives  a 
resultant  rn.m.f.  producing  the  instantaneous  value  of  field  flux. 
Immediately  after  short  circuit,  while  the  field  flux  still  has  full 
value,  that  is,  before  it  has  appreciably  decreased,  the  field  m.m.f. 
thus  must  have  increased  by  a  value  equal  to  the  counter  m.m.f. 
of  armature  reaction.  As  the  field  is  still  practically  unchanged, 
the  generated  e.m.f.  is  the  nominal  generated  voltage,  e0,  and 
the  short-circuit  current  is 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS 


201 


and  from  this  value  gradually  dies  down,  with  a  decrease  of  the 
field  flux  and  of  the  generated  e.m.f.,  to 


Hence,  approximately,  when   short-circuiting  an   alternator, 
in  the  first  moment  the  short-circuit  current  is 


x' 


while  the  field  current  has  increased  from  its  normal  value  i0  to 
the  value 

Field  excitation  +  Armature  reaction  m 

Field  excitation 
gradually  the  armature  current  decreases  to 


and  the  field  current  again  to  the  normal  value  i0. 

Therefore,  the  momentary  short-circuit  current  of  an  alternator 
bears  to  the  permanent  short-circuit  current  the  ratio 


,vo 


that  is, 


Armature  self-inductance  +  Armature  reaction 
Armature  self-inductance 


In  machines  of  high  self-inductance  and  low  armature  reaction, 
as  uni-tooth  high  frecfUency  alternators,  this  increase  of  the 
momentary  short-circuit  current  over  the  permanent  short- 
circuit  current  is  moderate,  but  may  reach  enormous  values  in 
machines  of  low  self-inductance  and  high  armature  reaction,  as 
large  low  frequency  turbo  alternators. 

114.  Superimposed  upon  this  transient  term,  resulting  from 
the  gradual  adjustment  of  the  field  flux  to  a  change  of  m.m.f.,  is 
the  transient  term  of  armature  reaction.  In  a  polyphase 
alternator,  the  resultant  m.m.f.  of  the  .armature  in  permanent 
conditions  is  constant  in  intensity  and  revolves  with  regard  to 
the  armature  at  uniform  synchronous  speed,  hence  is  stationary 


202 


TRANSIENT  PHENOMENA 


with  regard  to  the  field.  In  the  first  moment,  however,  the 
resultant  armature  m.m.f .  is  changing  in  intensity  and  in  velocity, 
approaching  its  constant  value  by  a  series  of  oscillations,  as 
discussed  in  Chapter  XIII.  Hence,  with  regard  to  the  field,  the 
transient  term  of  armature  reaction  is  pulsating  in  intensity  and 
oscillating  in  position,  and  therefore  generates  in  the  field  coils 


Field  Current 


Armature  Current 


Fig.  50.     Three-phase  short-circuit  current  of  a  turbo-alternator. 

an  e.m.f.  and  causes  a  corresponding  pulsation  in  the  field 
current  and  field  terminal  voltage,  of  the  same  frequency  as 
the  armature  current,  as  shown  by  the  oscillogram  of  such  a 
three-phase  short-circuit,  in  Fig.  50.  This  pulsation  of  field 
current  is  independent  of  the  point  in  the  wave,  at  which  the 
short-circuit  occurs,  and  dies  out  gradually,  with  the  dying  out 
of  the  transient  term  of  the  rotating  m.m.f. 

In  a  single-phase  alternator,  the  armature  reaction  is  alter- 
nating with  regard  to  the  armature,  hence  pulsating,  with  double 
frequency,  with  regard  to  the  field,  varying  between  zero  and  its 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS         203 

maximum  value,  and  therefore  generates  in  the  field  coils  a 
double  frequency  e.m.f.,  producing  a  pulsation  of  field  current 
of  double  frequency.  This  double-frequency  pulsation  of  the 
field  current  and  voltage  at  single-phase  short-circuit  is  pro- 
portional to  the  armature  current,  and  does  not  disappear 
with  the  disappearance  of  the  transient  term,  but  persists  also 
after  the  permanent  condition  of  short-circuit  has  been  reached, 


Armature 
current 


"Field 
current 


'  BHBnSSMMttHKattMMMNMBflUffiHMHMMHBB 
Fig.  61.     Single-phase  short-circuit  current  of  a  three-phase  turbo-alternator. 

merely  decreasing  with  the  decrease  of  the  armature  current. 
It  is  shown  in  the  oscillogram  of  a  single-phase  short-circuit  on 
a  three-phase  alternator,  Fig.  51. 

Superimposed  on  this  double  frequency  pulsation  is  a  single- 
frequency  pulsation  due  to  the  transient  term  of  the  armature 
current,  that  is,  the  same  as  on  polyphase  short-circuit.  With 
single-phase  short-circuit,  however,  this  normal  frequency  pul- 
sation of  the  field  depends  on  the  point  of  the  wave  at  which 
the  short-circuit  occurs,  and  is  zero,  if  the  circuit  is  closed  at 
the  moment  when  the  short-circuit  current  is  zero,  as  in  Fig.  51, 
and  a  maximum  when  the  short-circuit  starts  at  the  maximum 
point  of  the  current  wave.  As  this  normal  frequency  pulsation 
gradually  disappears,  it  causes  the  successive  waves  of  the 
double  frequency  pulsation  to  be  unequal  in  size  at  the 
beginning  of  the  transient  term,  and  gradually  become  equal, 
as  shown  in  the  oscillogram,  Fig.  52. 

The  calculation  of  the  transient  term  of  the  short-circuit 
current  of  alternators  thus  involves  the  transient  term  of  the 


204 


TRANSIENT  PHENOMENA 


armature  and  the  field  current,  as  determined  by  the  self- 
inductance  of  armature  and  of  field  circuit,  and  the  mutual 
inductance  between  the  armature  circuits  and  the  field  circuit, 
and  the  impressed  or  generated  voltage;  therefore  is  rather 
complicated;  but  a  simpler  approximate  calculation  can  be 


A 


Armature 
current 


Field 
current 


62.5  amp. 


Fig.  52.     Single-phase  short-circuit  current  of  a  three-phase  turbo-alternator. 

given  by  considering  that  the  duration  of  the  transient  term  is 
short  compared  with  that  of  the  armature  reaction  on  the  field. 

(A)  Polyphase  alternator. 

115.  Let  np  =  number  of  phases;  6  =  2nft  =  time-phase 
angle;  n0  =  number  of  field  turns  in  series  per  pole;  n^  number 
of  armature  turns  in  series  per  pole;  Z0  =  r0  —  jx0  =  self-inductive 
impedance  of  field  circuit;  Zl  =  rt  —  jxl  =  self-inductive  impe- 
dance of  armature  circuit;  p  =  permeance  of  field  magnetic  cir- 
cuit; a  =  2  7ifn1 10~8  =  induction  coefficient  of  armature;  E0  = 

Tji 

exciter  voltage;  /  0  =  — -  =  field  exciting  current,  in  permanent 

*• 

condition;  i0  =  field  exciting  current  at  time  0;  i'0°  =  field 
exciting  current  immediately  after  short-circuit;  i  =  armature 

2  re- 
current at  time  6,  and  kt  =  -     —  =  transformation  ratio  of  field 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS         205 

to  resultant  armature.  Counting  the  time  angle  6  from  the 
moment  of  short  circuit,  6  =  0,  and  letting  6'  =  time-phase 
angle  of  one  of  the  generator  circuits  at  the  moment  of  short 
circuit,  we  have, 

SF0  =  n0/0  =  field  excitation,  in  permanent  or  stationary  con- 
dition, (1) 

^o  =  P&o  =  PnJo  =  magnetic  flux  corresponding  thereto, 
and 

C  "  aP*o  =  aapnJ0.  (2) 

=  nominal  generated  voltage,  maximum  value,  at  0  =  0. 

Hence,  r 


=  momentary  short-circuit  current  at  time  0  —  0,  and 


=  resultant  armature  reaction  thereof. 

Assume  this  armature  reaction  as  opposite  to  the  field  excita- 
tion, 

37  -  Vo°;  (5) 

as  is  the  case  at  short  circuit. 

The  resultant  m.m.f.  of  the  magnetic  circuit  at  the  moment 
of  short-circuit  is 

£F0  -  £F0°  -  JFi0.  (6) 

At  this  moment,  however,  the  field  flux  is  still  <£0,  and  the  result- 
ant m.m.f.  is  given  by  (1)  as 

•!F0  =  JF0  =  n0/0.  (7) 

Substituting  (4),  (5),  (7)  in  (6)  gives 


hence,  ° 


, 

1 

0   =  - 


206  TRANSIENT  PHENOMENA 


Writing  *2=«,  (9) 


we  have  i0°  =  --Z-J70;  (10) 

that  is,  at  the  moment  of  short  circuit  the  field  exciting  current 
rises  from  70  to  i°,  and  then  gradually  dies  down  again  to  70  at 

r°  Q 

a  rate  depending  on  the  field  impedance  Z0,  that  is,  by  e    ^    ,  as 
discussed  in  preceding  chapters.     Hence,  it  can  be  represented 

by 


x, 

The  resultant  armature  m.m.f.,  or  armature  reaction,  is 

npnj° 

~2~ 

thus  the  magnetic  flux  which  would  be  produced  by  it  is 

pnpnj°  - 

~T~' 

and  therefore  the  voltage  generated  by  this  flux  is 

apnpnj° 
~2~ 

hence, 


Voltage  corresponding  to  the  m.m.f.  of  armature  current. 
Armature  current 

that  is,  x2  is  the  equivalent  or  effective  reactance  of  armature 
reaction. 

In  equations  (10)  and  (11)  the  external  self-inductance  of  the 
field  circuit,  that  is,  the  reactance  of  the  field  circuit  outside  of 
the  machine  field  winding,  has  been  neglected.  This  would 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS 


207 


introduce  a  negative  transient  term  in  (11),  thus  giving  equation 
(11)  the  approximate  form 


0-0       _ 


(12) 


where  #3  =  self-inductive  reactance  of  the  field  circuit  outside 
of  alternator  field  coils. 

The  more  complete  expression  requires  consideration  when 
x3  is  very  large,  as  when  an  external  reactive  coil  is  inserted  in 
the  field  circuit. 

In  reality,  x2  is  a  mutual  inductive  reactance,  and  x3  can  be 
represented  approximately  by  a  corresponding  increase  of  xr 

116.    If  /  =  maximum  value  of  armature  current,  we  have 


hence, 


and 


and 


npnj 

— — —  =  armature  m.m.f., 


.       npnj 

00  o 

=  resultant  m.m.f., 
E  =  ap& 

=  e.m.f.  maximum  generated  thereby, 


(13) 


(14) 


=  armature  current,  maximum. 

Substituting  (13)  in  (14)  gives 

npapn 
xj  = 


and 


or,  by  (9), 


n^apn^      x^  +  x2 
2    "  ' 


I=*n-*i, 


(15) 


(16) 


208 


TRANSIENT  PHENOMENA 


where    —  —  =  kt  =  transformation  ratio    of    field    turns  to 


n 


resultant  armature  turns;  hence, 

T  —  I"! 
i  —  h,ti( 


,  + 


(17) 


Substituting  (11)  in  (17)  thus  gives  the  maximum  value  of  the 
armature  current  as 


•TO 


9xt  +  x2  xl 

the  instantaneous  value  of  the  armature  current  as 


(18) 


i  =  ktl{ 


cos  (6  -.*)>-«    **   cos  V    ,    (19) 


X.  (X.   +  £,) 

1    v     1      '          2' 

and  by  equation  (10)  of  Chapter  XIII,  the  armature  reaction  as 


f 


-*9\ 

.  +  xjs     x°  )(  .         -^e         n) 
? ?-r L  <  I  -  e    *'    cos  0  f , 


(20) 


where  x^  +  x2  =  x0  is  the  synchronous  reactance  of  the  alter- 
nator. 

For  6  =  oo,  or  in  permanent  condition,  equations  (18),  (19), 
(20)  assume  the  usual  form: 


and 


i  =  kt!0  -2  cos  (0 

x 


f 

f 


(21) 


117.  As  an  example  is  shown,  in  Fig.  53,  the  instantaneous 
value  of  the  transient  short-circuit  current  of  a  three-phase 
alternator,  with  the  time  angle  6  as  abscissas,  and  for  the  con- 
stants: the  field  turns,  n0  =  100;  the  normal  field  current, 
70  =  200  amp.;  the  field  impedance,  Z0  =  rQ  -  jx0  =  1.28  - 
160  j  ohms;  the  armature  turns,  n^  =  25,  and  the  armature 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS          209 

impedance,    Zl  =  rl  -  jx^  -  0.4  -  5  /   ohms.     For  the   phase 
angle,  0'  =  0,  the  transformation  ratio  then  is 

np  nl       3 
and  the  equivalent  impedance  of  armature  reaction  is 


=  15, 
and  we  have 

7=       400(1  +  3s-0'008'),  (18) 

i  =       400  (1  +  3  £-°-008')  (cos  0  -  £-°-08'),       (19) 

and  /  =  15,000  (1  +  3  e-°-008')  (1  -  e'0'080  cos  0).     (20) 


1600 

800 

0 

-800 
-1600 


4Q06  +  3C 


\y 


!OS   0 


Fig.  53.     Short-circuit  current  of  a  three-phase  alternator. 

(B)  Single-phase  alternator. 

118.  In  a  single-phase  alternator,  or  in  a  polyphase  alternator 
with  one  phase  only  short-circuited,  the  armature  reaction  is 
pulsating. 

The  m.m.f.  of  the  armature  current, 

i  =  7  cos  (0  -  6'),  (22) 

of  a  single-phase  alternator,  is,  with  regard  to  the  field, 


hence,   for  position  angle  00  =  time  angle  0,  or  synchronous 
rotation, 

^=7(1  +  cos  2  (0-0')};  (23) 


210  TRANSIENT  PHENOMENA 

that  is,  of  double  frequency,  with  the  average  value, 


(24) 


pulsating  between  0  and  twice  the  average  value. 

The  average  value  (24)  is  the  same  as  the  value  of  the  poly- 
phase machine,  for  np  =  1. 

Using  the  same  denotations  as  in  (A),  we  have: 

(1)    JF0   =  Vo,  (25) 

(26) 
Denoting  the  effective  reactance  of  armature  reaction  thus : 

x2=^JT>  (27) 

and  substituting  (27)  in  (26)  we  obtain 

fri°  =-Vofl  +  cos  2  (0  -  0')}  =  -X/o  {1  + cos  20'};    (28) 
xl  xl 

hence,  by  (6), 

Vo  =  Vo°  -  -XU1  +  cos  2  6'} 
i 

and 

t'0°  =  — -2/0  \\  H — —  cos  2  Q'  i  ,  (29) 

and  the  field  current, 

r°  a 

X     -f  X  £~*»  (  X  ) 

xl  x^  +  a:2 

119.   If  /  =  maximum  value  of  armature  current, 

aFt  =|'/  {1  +  cos  2(0  -0')}  (31) 

=  armature  m.m.f.; 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS          211 

hence, 

SF  =  Vo  -  fri  (32) 

=  resultant  m.m.f. 
Since,  however, 


(33) 
*i       •*•! 

and,  by  (27), 


we  have,  by  (33) 


17  =  7.  (34) 

ap          2  x2 


Substituting  (30),  (31),  and  (34)  into  (32)  gives 


7  =  n 
2  x 


or,  substituting, 

kt  =  2  —  °  =  transformation  ratio,  (35) 

ni 

and  rearranging,  gives 

'  -  *^  "j±?l!: 

as  the  maximum  value  of  the  armature  current. 

This  is  the  same  expression  as  found  in  (18)  for  the  poly- 
phase machine,  except  that  now  the  reactances  have  different 
values. 


212 


TRANSIENT  PHENOMENA 


Heref rom  it  follows  that  the  instantaneous  value  of  the  armature 
current  is 

~  *°6)    (  -10  ) 

'  !  cos  (0  -  6')  -  e    **    cos  6'  {  ,    (37) 


and,  by  (31),  the  armature  reaction  is 


(38) 


For  0  =  oo,  or  permanent  condition,  equations  (30),  (36),  (37), 
and  (38)  give 


v-VU* 


cos  2  (0  -  00 1 , 


X,  +  X, 


cos  (0  -  00, 


and 


cos2(0-00}. 


(39) 


As  seen,  the  field  current  i0  is  pulsating  even  in  permanent 
condition,  the  more  so  the  higher  the  armature  reaction  x2 
compared  with  the  armature  self-inductive  reactance  xr 

120.  Choosing  the  same  example  as  in  Fig.  52,  paragraph 
117,  but  assuming  only  one  phase  short-circuited,  that  is,  a  single- 
phase  short  circuit  between  two  terminals,  we  have  the  effective 
armature  series  turns,  nv  =  25  V3  =  43.3;  the  armature  impe- 
dance, Zl  =  rl  -  jx^  =  0.8  —  10 /;  6'  =  0;  the  transformation 
ratio,  kt  =  4.62,  and  the  effective  reactance  of  armature  reaction 

o 

x2  =  —x0  =15;  herefrom, 

7  =  555(1  +  1.5e-°-op8'),  (36) 

i  =  555  (1  +  1.5  £-°-008')  (cos  0  -  r*'080),       (37) 

and  /  -  12,000  (1  +  1.5£-°-008fl)  (1  +  cos  2  0);       (38) 


SHORT-CIRCUIT  CURRENTS  OF  ALTERNATORS          213 

and  the  field  current  is 

iQ  =  200  (1  +  1.5  £-°-008 ' )  (1  +  0.6  cos  2  6).  (30) 

In  this  case,  in  the  open-circuited  phase  of  the  machine,  a 
high  third  harmonic  voltage  is  generated  by  the  double  frequency 
pulsation  of  the  field,  and  to  some  extent  also  appears  in  the 
short-circuit  current. 


SECTION  II 
PERIODIC   TRANSIENTS 


PEKIODIC    TRANSIENTS 


CHAPTER  I. 

INTRODUCTION. 

1.  Whenever  in  an  electric  circuit  a  sudden  change  of  the 
circuit  conditions  is  produced,  a  transient  term  appears  in  the 
circuit,  that  is,  at  the  moment  when  the  change  begins, 
the  circuit  quantities,  as  current,  voltage,  magnetic  flux, etc.,  cor- 
respond to  the  circuit  conditions  existing  before  the  change,  but 
do  not,  in  general,  correspond  to  the  circuit  conditions  brought 
about  by  the  change,  and  therefore  must  pass  from  the  values 
corresponding  to  the  previous  condition  to  the  values  corre- 
sponding to  the  changed  condition.  This  transient  term  may  be 
a  gradual  approach  to  the  final  condition,  or  an  approach  by  a 
series  of  oscillations  of  gradual  decreasing  intensities. 

Gradually  —  after  indefinite  time  theoretically,  after  relatively 
short  time  practically  —  the  transient  term  disappears,  and 
permanent  conditions  of  current,  of  voltage,  of  magnetism,  etc., 
are  established.  The  numerical  values  of  current,  of  voltage,  etc., 
in  the  permanent  state  reached  after  the  change  of  circuit  con- 
ditions, in  general,  are  different  from  the  values  of  current, 
voltage,  etc.,  existing  in  the  permanent  state  before  the  change, 
since  they  correspond  to  a  changed  condition  of  the  circuit. 
They  may,  however,  be  the  same,  or  such  as  can  be  considered 
the  same,  if  the  change  which  gives  rise  to  the  transient  term 
can  be  considered  as  not  changing  the  permanent  circuit  con- 
ditions. For  instance,  if  the  connection  of  one  part  of  a  circuit, 
with  regard  to  the  other  part  of  the  circuit,  is  reversed,  a  transient 
term  is  produced  by  this  reversal,  but  the  final  or  permanent 
condition  after  the  reversal  is  the  same  as  before,  except  that 
the  current,  voltage,  etc.,  in  the  part  of  the  circuit  which  has  been 
reversed,  are  now  in  opposite  direction.  In  this  latter  case, 
the  same  change  can  be  produced  again  and  again  after  equal 

217 


218  TRANSIENT  PHENOMENA 

intervals  of  time  t0,  and  thus  the  transient  term  made  to  recur 
periodically.  The  electric  quantities  i,  e,  etc.,  of  the  circuit, 
from  time  t  =  0  to  t  =  tQ,  have  the  same  values  as  from  time 
t  =  t0  to  t  =  2  tof  from  t  =  2  t0  to  t  =  3  tot  etc.,  and  it  is  sufficient 
to  investigate  one  cycle,  from  t  =  0  to  t  =  t0. 

In  this  case,  the  starting  values  of  the  electrical  quantities 
during  each  period  are  the  end  values  of  the  preceding  period, 
or,  in  other  words,  the  terminal  values  at  the  moment  of  start 
of  the  transient  term,  t  =  0,  i  =  i0  and  e  =  e0,  are  the  same  as 
the  values  at  the  end  of  the  period  t  =  t0,  i  =  i'  and  e  =  e'\ 
that  is,  iQ  =  ±  i' j  e0  =  ±  e',  etc. ;  where,  the  plus  sign  applies 
for  the  unchanged,  and  the  minus  sign  for  the  reversed  part  of  the 
circuit. 

2.  With  such  periodically  recurrent  changes  of  circuit  con- 
ditions, the  period  of  recurrence  t0  may  be  so  long,  that  the 
transient  term  produced  by  a  change  has  died  out,  the  permanent 
conditions  reached,  before  the  next  change  takes  place.     Or, 
at  the  moment  where  a  change  of  circuit  conditions  starts,  a 
transient  term,  the  transient  term  due  to  the  preceding  change, 
has  not  yet  disappeared,  that  is,  the  time,  t0,  of  a  period  is  shorter 
than  the  duration  of  the  transient  term. 

In  the  first  case,  the  terminal  or  starting  values,  that  is,  the 
values  at  the  moment  when  the  change  begins,  are  the  same  as 
the  permanent  values,  and  periodic  recurrence  has  no  effect  on 
the  character  of  the  transient  term,  but  the  phenomenon  is  cal- 
culated as  discussed  in  Section  I,  as  single  transient  term, 
which  gradually  dies  out. 

If,  however,  at  the  moment  of  change,  the  transient  term  of 
the  preceding  change  has  not  yet  vanished,  then  the  starting  or 
terminal  values  of  the  electric  quantities,  as  i0  and  e0,  also  contain 
a  transient  term,  namely,  that  existing  at  the  end  of  the  preced- 
ing period.  The  same  term  then  exists  also  at  the  end  of  the 
period,  or  at  t  =  t0.  Hence  in  this  case,  the  terminal  conditions 
are  given,  not  as  fixed  numerical  values,  but  as  an  equation 
between  the  electric  quantities  at  time  t  =  0  and  at  time  t  =  t0; 
or,  at  the  beginning  and  at  the  end  of  the  period,  and  the  inte- 
gration constants,  thus,  are  calculated  from  this  equation. 

3.  In  general,  the  permanent  values  of  electric  quantities 
after  a  change  are  not  the  same  as  before,  and  therefore  at  least 
two  changes  are  required  before  the  initial  condition  of  the 


INTRODUCTION  219 

circuit  is  restored,  and  the  cycle  can  be  repeated.  Periodically 
recurring  transient  phenomena,  thus  usually  consist  of  two  or 
more  successive  changes,  at  the  end  of  which  the  original  con- 
dition of  the  circuit  is  reproduced,  and  therefore  the  series  of 
changes  can  be  repeated.  For  instance,  increasing  the  resistance 
of  a  circuit  brings  about  a  change.  Decreasing  this  resistance 
again  to  its  original  value  brings  about  a  second  change,  which 
restores  the  condition  existing  before  the  first  change,  and  thus 
completes  the  cycle.  In  this  case,  then,  the  starting  values  of 
the  electric  quantities  during  the  first  part  of  the  period  equal 
the  end  values  during  the  second  part  of  the  period,  and  the 
starting  values  of  the  second  part  of  the  period  equal  the  end 
values  of  the  first  part  of  the  period.  That  is,  if  a  resistor  is 
inserted  at  time  t  =  0,  short  circuited  at  time  t  =  t1}  and  inserted 
again  at  time  t  =  t0,  and  e  and  i  are  voltage  and  current  respec- 
tively during  the  first,  e1  and  il  during  the  second  part  of  the 
period,  we  have 

/e/t,Q  =  A!/^;     A,/,=<1  -  /e/t=k, 
and 

AA=o  =  AiA-ioJ  Ai/«=«,  =  AA=«i- 

If  during  the  times  ^  and  t0  -  tt  the  transient  terms  have 
already  vanished,  and  permanent  conditions  established,  so  that 
the  transient  terms  of  each  part  of  the  period  depend  only  upon 
the  permanent  values  during  the  other  part  of  the  period,  the 
length  of  time  tl  and  t0  has  no  effect  on  the  transient  term,  that 
is,  each  change  of  circuit  conditions  takes  place  and  is  calculated 
independently  of  the  other  change,  or  the  periodic  recurrence. 
A  number  of  such  cases  have  been  discussed  in  Section  I,  as 
for  instance,  the  effect  of  cutting  a  resistor  in  and  out  of  a 
divided  inductive  circuit,  paragraph  75,  Fig.  33.  In  this  case, 
four  successive  changes  are  made  before  the  cycle  recurs:  a 
resistor  is  cut  in,  in  two  steps,  and  cut  out  again  in  two 
steps,  but  at  each  change,  sufficient  time  elapses  to  reach 
practically  permanent  condition. 

In  general,  and  especially  in  those  cases  of  periodic  transient 
phenomena,  which  are  of  engineering  importance,  successive 
changes  occur  before  the  permanent  condition  is  reached,  or 
even  approximated  after  the  preceding  change,  so  that  frequently 


220  TRANSIENT  PHENOMENA 

the  values  of  the  electric  quantities  are  very  different  throughout 
the  whole  cycle  from  the  permanent  values  which  they  would 
gradually  assume;  that  is,  the  transient  term  preponderates 
in  the  values  of  current,  voltage,  etc.,  and  the  permanent  term 
occasionally  is  very  small  compared  with  the  transient  term. 

4.  Periodic  transient  phenomena  are  of  engineering  impor- 
tance mainly  in  three  cases:  (1)  in  the  control  of  electric  circuits; 
(2)  in  the  production  of  high  frequency  currents,  and  (3)  in  the 
rectification  of  alternating  currents. 

1.  In  controlling  electric  circuits,  etc.,  by  some  operating 
mechanism,  as  a  potential  magnet  increasing  and  decreasing  the 
resistance  of  the  circuit,  or  a  clutch  shifting  brushes,  etc.,  the 
main  objections  are  due  to  the  excess  of  the  friction  of  rest  over 
the  friction  while  moving.     This  results  in  a  lack  of  sensitiveness, 
and  an  overreaching  of  the  controlling  device.     To  overcome 
the  friction  of  rest,  the  deviation  of  the  circuit  from  normal 
must  become  greater  than  necessary  to  maintain  the  motion  of 
the  operating  mechanism,,  and  when  once  started,  the  mechanism 
overreaches.    This  objection  is  eliminated  by  never  allowing 
the  operating  mechanism  to  come  to  rest,  but  arranging  it  in 
unstable  equilibrium,  as  a  " floating  system,"  so  that  the  con- 
dition of  the  circuit  is  never  normal,   but  continuously  and 
periodically  varies  between  the  two  extremes,  and  the  resultant 
effect  is  the  average  of  the  transient  terms,  which  rapidly  and 
periodically   succeed    each    other.     By    changing   the    relative 
duration  of  the  successive  transient  terms,  any  resultant  inter- 
mediary between  the  two  extremes  can  thus  be  produced.     On 
this  principle,  for  instance,  operated  the  controlling  solenoid  of 
the  Thomson-Houston  arc  machine,  and  also  numerous  auto- 
matic potential  regulators. 

2.  Production  of  high  frequency  oscillating  currents  by  period- 
ically recurring  condenser  discharges  has  been  discussed  under 
" oscillating  current  generator,"  in  Section  I,  paragraph  44. 

Non-sinusoidal  high  frequency  alternating  currents  are  pro- 
duced by  an  arc,  when  made  unstable  by  shunting  it  with  a 
condenser,  as  discussed  before. 

The  Ruhmkorff  coil  or  inductorium  also  represents  an  appli- 
cation of  periodically  recurring  transient  phenomena,  as  also 
does  Prof.  E.  Thomson's  dynamostatic  machine. 

3.  By  reversing  the  connections  between  a  source  of  alter- 


INTRODUCTION  221 

nating  voltage  and  the  receiver  circuit,  synchronously  with  the 
alternations  of  the  voltage,  the  current  in  the  receiver  circuit  is 
made  unidirectional  (though  more  or  less  pulsating)  and  there- 
fore rectified. 

In  rectifying  alternating  voltages,  either  both  half  waves  of 
voltage  can  be  taken  from  the  same  source,  as  the  same  trans- 
former coil,  and  by  synchronous  reversal  of  connections  sent  in 
the  same  direction  into  the  receiver  circuit,  or  two  sources  of 
voltage,  as  the  two  secondary  coils  of  a  transformer,  may  be 
used,  and  the  one  half  wave  taken  from  the  one  source,  and  sent 
into  the  receiver  circuit,  the  other  half  wave  taken  from  the 
other  source,  and  sent  into  the  receiver  circuit  in  the  same 
direction  as  the  first  half  wave.  The  latter  arrangement  has 
the  disadvantage  of  using  the  alternating  current  supply  source 
less  economically,  but  has  the  advantage  that  no  reversal,  but 
only  an  opening  and  closing  of  connections,  is  required,  and  is 
therefore  the  method  commonly  applied  in  stationary  rectify- 
ing apparatus. 

6.  In  rectifying  alternating  voltages,  the  change  of  connec- 
tions between  the  alternating  supply  and  the  unidirectional 
receiving  circuit  can  be  carried  out  as  outlined  below : 

(a)  By  a  synchronously  moving  commutator  or  contact 
maker,  in  mechanical  rectification.  Such  mechanical  rectifiers 
may  again  be  divided,  by  the  character  of  the  alternating  supply 
voltage,  into  single  phase,  quarter  phase  and  three  phase,  and 
by  the  character  of  the  electric  circuit,  into  constant  potential 
and  constant  current  rectifiers.  Mechanical  rectification  by  a 
commutator  driven  by  a  separate  synchronous  motor  has  not  yet 
found  any  industrial  application.  Rectification  by  a  commuta- 
tor driven  by  the  generator  of  the  alternating  voltage  has  found 
very  extended  and  important  industrial  use  in  the  excitation  of 
the  field,  or  a  part  of  the  field  (the  series  field)  of  alternators  and 
synchronous  motors,  and  especially  in  the  constant-current  arc 
machine.  The  Brush  arc  machine  is  a  quarter-phase  alternator 
connected  to  a  rectifying  commutator  on  the  armature  shaft, 
and  the  Thomson-Houston  arc  machine  is  a  star-connected 
three-phase  alternator  connected  to  a  rectifying  commutator  on 
the  armature  shaft.  The  reason  for  using  rectification  in  these 
machines,  which  are  intended  to  produce  constant  direct  current 
at  very  high  voltage,  is  that  the  ordinary  commutator  of  the 


222  TRANSIENT  PHENOMENA 

continuous-current  machine  cannot  safely  commutate,  even  at 
limited  current,  more  than  30  to  50  volts  per  commutator 
segment,  while  the  rectifying  commutator  of  the  constant- 
current  arc  machine  can  control  from  2000  to  3000  volts  per 
segment,  and  therefore  rectification  is  superior  to  commutation 
for  very  high  voltages  at  limited  current,  as  explained  by  the 
character  of  this  phenomenon,  discussed  in  Chapter  III. 

(b)  The  synchronous  change  of  circuit  connection  required 
by  the  rectification  of  alternating  e.m.fs.  can  be  brought  about 
without  any  mechanical  motion  in  so-called   "arc  rectifiers," 
by  the  characteristic  properties  of  the  electric  arc,  to  be  a  good 
conductor  in  one,  an  insulator  in  the  opposite  direction.     By 
thus  inserting  an  arc  in  the  path  of  the  alternating  circuit, 
current  can  exist  and  thus  a  circuit  be  established  for  that  half 
wave  of  alternating  voltage,  which  sends  the  current  in  the 
same  direction  as  the  current  in  the  arc,  while  for  the  reversed 
half  wave  of  voltage  the  arc  acts  as  open  circuit.     As  seen,  the 
arc  cannot  reverse,  but  only  open  and  close  the  circuit,  and  so 
can  rectify  only  one  half  wave,  that  is,  two  separate  sources  of 
alternating  voltage,  or  two  rectifiers  with  the  same  source  of 
voltage,  are  required  to  rectify  both  half  waves  of  alternating 
voltage. 

(c)  Some  electrolytic  cells,  as  those  containing  aluminum  as 
one  terminal,  offer  a  low  resistance  to  the  passage  of  current  in 
one  direction,  but  a  very  high  resistance,  or  practically  interrupt 
the  current,  in  opposite  direction,  due  to  the  formation  of  a  non- 
conducting film  on  the  aluminum,  when  it  is  the  positive  terminal. 
Such  electrolytic  cells  can  therefore  be  used  for  rectification  in 
a  similar  manner  as  arcs. 

The  three  main  classes  of  rectifiers  thus  are:  (a)  mechanical 
rectifiers;  (b)  arc  rectifiers;  (c)  electrolytic  rectifiers. 

Still  other  methods  of  rectification,  as  by  the  unidirectional 
character  of  vacuum  discharges,  of  the  conduction  in  some 
crystals,  etc.,  are  not  yet  of  industrial  importance. 


CHAPTER  II. 

CIRCUIT   CONTROL   BY    PERIODIC  TRANSIENT    PHENOMENA. 

6.  As  an  example  of  a  system  of  periodic  transient  phenomena, 
used  for  the  control  of  electric  circuits,  may  be  considered  an 
automatic  potential  regulator  operating  in  the  field  circuit  of 
the  exciter  of  an  alternating  current  system. 

Let,  r0  =  40  ohms  =  resistance  and  L  =  400  henrys  = 
inductance  of  the  exciter  field  circuit. 

A  resistor,  having  a  resistance,  rl  =  24  ohms,  is  inserted  in 
series  to  r0,  L  in  the  exciter  field,  and  a  potential  magnet,  con- 
trolled by  the  alternating  current  system,  is  arranged  so  as  to 
short  circuit  resistance,  rv  if  the  alternating  potential  is  below, 
to  throw  resistance  rl  into  circuit  again,  if  the  potential  is 
above  normal. 

With  a  single  resistance  step,  rv  in  the  one  position  of  the 
regulator,  with  rx  short  circuited,  and  only  r0  as  exciter  field 
winding  resistance,  the  alternating  potential  would  be  above 
normal,  that  is,  the  regulator  cannot  remain  in  this  position, 
but  as  soon  after  short  circuiting  resistance  rl  as  the  potential 
has  risen  sufficiently,  the  regulator  must  change  its  position 
and  cut  resistance  T\  into  the  circuit,  increasing  the  exciter  field 
circuit  resistance  to  r0  +  rr  This  resistance  now  is  too  high, 
would  lower  the  alternating  potential  too  much,  and  the  regula- 
tor thus  cuts  resistance  rl  out  again.  That  is,  the  regulator 
continuously  oscillates  between  the  two  positions,  corresponding 
to  the  exciter  field  circuit  resistances  r0  and  (r0  +  rt)  respec- 
tively, at  a  period  depending  on  the  momentum  of  the  moving 
mass,  the  force  of  the  magnets,  etc.,  that  is,  approximately 
constant.  The  time  of  contact  in  each  of  the  two  positions, 
however,  varies:  when  requiring  a  high  field  excitation,  the 
regulator  remains  a  longer  time  in  position  r0,  hence  a  shorter 
time  in  position  (r0  +  rt),  before  the  rising  potential  throws  it 
over  into  the  next  position;  while  at  light  load,  requiring  low 
field  excitation,  the  duration  of  the  period  of  high  resistance, 

223 


224 


TRANSIENT   PHENOMENA 


(TO  _|_  rj}  is  greater,  and  that  of  the  period  of  low  resistance,  r0, 
less. 

7.  Let,  ^  =  the  duration  of  the  short  circuit  of  resistance  rx; 
t2  =  the  time  during  which  resistance  rx  is  in  circuit,  and  t0  = 

t,  +  tr 

During  each  period  t0,  the  resistance  of  the  exciter  field, 
therefore,  is  r0  for  the  time  tv  and  (r0  +  rj  for  the  time  ty 

Furthermore,  let,  i1  =  the  current  during  time  tv  and  i2  = 
the  current  during  time  tr 

During  each  of  the  two  periods,  let  the  time  be  counted 
anew  from  zero,  that  is,  the  transient  current  il  exists  during  the 
time  0  <  t  <  tv  through  the  resistance  r0,  the  transient 
current,  iv  during  the  time  0  <  t  <  t2,  through  the  resistance 
<>o  +  rj. 

This  gives  the  terminal  conditions : 


and 


(1) 


that  is,  the  starting  point  of  the  current,  iv  is  the  end  value  of 
the  current,  iv  and  inversely. 

If  now,  e  =  voltage  impressed  upon  the  exciter  field  circuit, 
the  differential  equations  are : 


and 


e  =  (T, 


di. 


(2) 


or, 


dil 


rt        -, 

ro 


dt. 


(3) 


CIRCUIT  CONTROL 


225 


Integrated, 


and 


ro  + 


(4) 


Substituting  the  terminal  conditions   (1)  in  equations    (4), 
gives  for  the  integration  constants  cl  and  c2  the  equations, 

e  e 


+   'I 


and 


herefrom, 


and 


ct  =  - 


(ro 


»"o 
-  T 

-      L 


erl-e 


(5) 


Substituting  (5)  in  (4), 


and 


1  + 


_  rp+ri 
L 


(6) 


If,  e  =  250  volts;  tQ  =  0.2  sec.,  or  5  complete  cycles  per  sec.; 
\  =  0.15,  and  t2  =  0.05  sec.;  then 

i,  =  6.25  {1  -  0.128  r°-li\ 


and 


i    =  3.91    1  +  0.391  fi 


-°-16' 


(7) 


226  TRANSIENT  PHENOMENA 

8.  The  mean  value  of  current  in  the  circuit  is 

1       (    Ck 
I'-T-rr    J  VB 

h  +  t2  '^o 

This  integrated  gives, 


(8) 


r0  +  rlt 


(9) 


and,  if 
and 


(10) 


are  the  two  extreme  values  of  permanent  current,  corresponding 
respectively  to  the  resistances  r0  and  (r0  +  rj,  we  have 


(11) 


that  is,  the  current,  i,  varies  between  i/  and  i2'  as  linear  function 
of  the  durations  of  contact,  ^  and  ty 

The  maximum  variation  of  current  during  the  periodic  change 
is  given  by  the  ratio  of  maximum  current  and  minimum  current; 
or, 


and  is 


where, 


and 


r0(l  -e-" 


r0(l  - 


(12) 


(13) 


(14) 


CIRCUIT  CONTROL  227 

Substituting 

x2       x3 
1-.        =,__  +  __+...,  (15) 

by  using  only  term  of  first  order; 

gives  (16) 

q  =  1;    J 

that  is,  the  primary  terms  eliminate,  and  the  difference  between 
it  and  i2  is  due  to  terms  of  secondary  order  only,  hence  very 
small. 

Substituting 

1  -".•"-*-!_;  (17) 

that  is,  using  also  terms  of  second  order,  gives 

(18) 

or,  approximately, 

q  =1  +r          r^2     rs    >  (19) 

and,  substituting  (14), 

g-i  +  LJ1*+O  ;  (2°) 

that  is,  the  percentage  variation  of  current  is 


Equation  (21)  is  a  maximum  for 


and,  then,  is 

9-1  =       ;  (23) 


228  TRANSIENT  PHENOMENA 

or,  in  the  above  example,      (rl  =  24;  L  =  400;  tQ  =  0.2); 

q  -  1  =  0.003; 

that  is,  0.3  per  cent. 

The  time  t0  of  a  cycle,  which  gives  1  per  cent  variation  of 
current,  q  —  1  =  0.01,  is 

t0  =  ^  (?  -  1),  (24) 

'  1 

=  f  sec. 

The  pulsation  of  current,  0.3  per  cent  respectively  1  per  cent, 
thus  is  very  small  compared  with  the  pulsation  of  the  resistance, 
rt  =  24  ohms,  which  is  46  per  cent  of  the  average  resistance 

r0  -f  ~  =  52  ohms. 


CHAPTER  III. 

MECHANICAL  RECTIFICATION. 

9.  If  an  alternating-current  circuit  is  connected,  by  means 
of  a  synchronously  operated  circuit  breaker  or  rectifier,  with  a 
second  circuit  in  such  a  manner,  that  the  connection  between 
the  two  circuits  is  reversed  at  or  near  the  moment  when  the 
alternating  voltage  passes  zero,  then  in  the  second  circuit 
current  and  voltage  are  more  or  less  unidirectional,  although 
they  may  not  be  constant,  but  pulsating. 

If  i  =  instantaneous  value  of  alternating  current,  and  i0  = 
instantaneous  value  of  rectified  current,  then  we  have,  before 
reversal,  i0  =  i,  and  after  reversal,  i0  =  —  i\  that  is,  during 
the  reversal  of  the  circuit  one  of  the  currents  must  reverse. 
Since,  however,  due  to  the  self-inductance  of  the  circuits,  neither 
current  can  reverse  instantly,  the  reversal  occurs  gradually, 
so  that  for  a  while  during  rectification  the  instantaneous  value 
of  the  alternating  and  of  the  rectified  current  differ  from  each 
other.  Thus  means  have  to  be  provided  either  to  shunt  the 
difference  between  the  two  currents  through  a  non-inductive 
bypath,  or,  the  difference  of  the  two  currents  exists  as  arc  over 
the  surface  of  the  rectifying  commutator.* 

The  general  phenomenon  of  single-phase  rectification  thus 
is :  The  alternating  and  the  rectified  circuit  are  in  series.  Both 
circuits  are  closed  upon  themselves  at  the  rectifier,  by  the 
resistances,  r  and  r0,  respectively.  The  terminals  are  reversed. 
The  shunt-resistance  circuits  are  opened,  leaving  the  circuits 
in  series  in  opposite  direction. 

Special  cases  hereof  are: 

1.  If  r  =  r0  =  0,  that  is,  during  rectification  both  circuits  are 
short  circuited.  Such  short-circuit  rectification  is  feasible  only 
in  limited-current  circuits,  as  on  arc  lighting  machines,  or  in 

*  If  the  circuit  is  reversed  at  the  moment  when  the  alternating  current 
passes  zero,  due  to  self-inductance  of  the  rectified  circuit  its  current  differs 
from  zero,  and  an  arc  still  appears  at  the  rectifier. 

229 


230  TRANSIENT  PHENOMENA 

cases  where  the  voltage  of  the  rectified  circuit  is  only  a  small  part 
of  the  total  voltage,  and  thus  the  current  not  controlled  thereby, 
as  when  rectifying  for  the  supply  of  series  fields  of  alternators. 

2.  r  =  r0  =  oo ,  or  open  circuit  rectification.  This  is  feasible 
only  if  the  rectified  circuit  contains  practically  no  self -inductance, 
but  a  constant  counter  e.m.f.,  e,  (charging  storage  batteries), 
so  that  in  the  moment  when  the  alternating  impressed  e.m.f. 
falls  to  e,  and  the  current  disappears,  the  circuit  is  opened,  and 
closed  again  in  opposite  direction  when  after  reversal  the  alter- 
nating impressed  e.m.f.  has  reached  the  value,  e. 

In  polyphase  rectification,  the  rectified  circuit  may  be  fed 
successively  by  the  successive  phases  of  the  system,  that  is 
shifted  over  from  a  phase  of  falling  e.m.f.  to  a  phase  of  rising 
e.m.f.,  by  shunting  the  two  phases  with  each  other  during  the 
time  the  current  changes  from  the  one  to  the  next  phase.  Thus 
the  Thomson-Houston  arc  machine  is  a  star-connected  three- 
phase  constant-current  alternator  with  rectifying  commutator. 
The  Brush  arc  machine  is  a  quarter-phase  machine  with  rectify- 
ing commutator. 

In  rectification  frequently  the  sine  wave  term  of  the  current 
is  entirely  overshadowed  by  the  transient  exponential  term, 
and  thus  the  current  in  the  rectified  circuit  is  essentially  of  an 
exponential  nature. 

As  examples,  three  cases  will  be  discussed: 

1.  Single-phase    constant-current    rectification;    that    is,    a 
rectifier  is  inserted  in  an  alternating-current  circuit,  and   the 
voltage  consumed  by  the  rectified  circuit  is  small  compared  with 
the  total  circuit  voltage;  the  current  thus  is  not  noticeably 
affected  by  the  rectifier.     In  other  words,  a  sine  wave  of  current 
is  sent  over  a  rectifying  commutator. 

2.  Single-phase   constant-potential   rectification;   that  is,    a 
constant-potential  alternating  e.m.f.  is  rectified,  and  the  impe- 
dance between  the  alternating  voltage  and  the  rectifying  com- 
mutator is  small,  so  that  the  rectified  circuit  determines  the 
current  wave  shape. 

3.  Quarter-phase  constant-current  rectification  as  occurring 
in  the  Brush  arc  machine. 


MECHA  NIC  A  L  REG  TIFICA  TION 


231 


i.   Single-phase  constant-current  rectification. 

10.  A  sine  wave  of  current,  i0  sin  0,  derived  from  an  e.m.f. 
very  large  compared  with  the  voltage  consumed  in  the  recti- 
fied circuit,  feeds,  after  rectification, 
a  circuit  of  impedance  Z  =  r  —  jx. 
This  circuit  is  permanently  shunted 
by  a  circuit  of  resistance  rr 

Rectification  takes  place  over  short- 
circuit  from  the  moment  n  —  02  to 
TT  +  0jj  that  is,  at  n  -  02the  rectified 
and  the  alternating  circuit  are  closed 
upon  themselves  at  the  rectifier,  and 
this  short-circuit  opened,  after  rever- 
sal, at  TT  +  6lf  as  shown  by  the  dia- 
grammatic representation  of  a  two- 
pole  model  of  such  a  rectifier  in  Fig. 
54.  In  this  case  the  space  angles 
TT  -f  TJ  and  TT  —  r2  and  the  time  angles 
TT  -f  Ol  and  TT  -  02  are  identical. 

This  represents  the  conditions  ex- 
isting   in     compound-wound    alter- 
nators,   that    is,    alternators    feeding    a    series    field    winding 
through  a  rectifier. 

Let,  during  the   period  from  Ol  to  n  -  02,  i  =  current  in 
impedance  Z,  and  il  =  current  in  resistance  rlt  then: 

i  +  i1  =  i0  sin  0. 
However, 

di 


Fig.  54.     Single-phase  current 
rectifier  commutator. 


(1) 


^1r1=^r 


(2) 


and  substituting  (1)  in  (2)  gives  the  differential  equation : 

di 

i  (r  +r1)  +  x~  -  i0rl  sin  0  =  0, 
au 


(3) 
(4) 


which  is  integrated  by  the  function : 

i  =  Ae-ae+  Bsin  (6  -  8). 
Substituting  (4)  in  (3)  and  arranging,  gives : 
A  (r  +  rl  -  ax)  e~  a&  +  [B  ( [r  +  rj  cos  d  +  x  sin  8)  -  i/J  sin  0 
-  [(r  +  rj  sin  d  -  x  cos  d]  B  cos  0  =  0,     (5) 


232  TRANSIENT  PHENOMENA 

which  equation  must  be  an  identity,  thus : 


and 

and  herefrom: 


r  +  rl  —  ax  =  0, 

B  ( [r  +  rj  cos  d  +  x  sin  d)  -  i0rl  =  0 
(r  +  PJL)  sin  d  —  x  cos  d  =  0, 


tand  = 


and 
where 

hence: 


r+r, 
B  =  i 


°  V(r+  rj' 


(6) 


z  =  V(r  +  r,)2  +  x2; 


(7) 


(8) 


During  the  time  of  short-circuit,  from  TT  —  02  to  TT  +  0t,  if 
i'  =  current  in  impedance  Z,  we  have 


di' 


hence: 


(9) 


(10) 


The  condition  of  sparkless  rectification  is,  that  no  sudden 
change  of  current  occur  anywhere  in  the  system.  In  consequence 
hereof  we  must  have : 

i  =  i'  =  i0  sin  0  at  the  moment  6  =  x  —  62, 

and,  at  the  moment  6  =  n  +  6V  i'  must  have  reached  the  same 
value  as  i  and  i0  sin  0  at  the  moment  0  =  0r 


MECHANICAL  RECTIFICATION  233 

This  gives  the  two  double  equations : 

and  (11) 


or,  substituting  (8)  and  (9), 

-  r  +  ri  (IT  -^         r  -  -  (*  -  *2) 

A  e      *  +  ^o-1  sin  (d  +  0a)  =  A'e    *  =  10  sin  ^2  (12) 

z 

and 


^  £  -  i0  -i  sin  0  -  OJ  =  A'e  =  i0  sin  <9r       (13) 

These  four  equations  (12)  (13)  determine  four  of  the  five 
quantities,  A,  A'  ,  Ov  02,  rv  leaving  one  indeterminate. 

Thus,  one  of  these  five  quantities  can  be  chosen.  The  deter- 
mination of  the  four  remaining  quantities,  however,  is  rather 
difficult,  due  to  the  complex  character  of  equations  (12)  (13), 
and  is  feasible  only  by  approximation,  in  a  numerical  example. 

11.  EXAMPLE:  Let  an  alternating  current  of  effective  value 
of  100  amp.,  that  is,  of  maximum  value  t'0  =  141.4,  be  rectified 
for  the  supply  of  a  circuit  of  impedance  Z  =  0.2  -  2  j,  shunted 
by  a  non-inductive  circuit  of  resistance  rr 

Let  the  series  connection  of  the  rectified  and  alternating 
circuits  be  established  30  time-degrees  after  the  zero  value  of 

alternating  current,  that  is,  d1  =  30  deg.  =  -  chosen. 
Then,  from  equation  (13),  we  have 

A'e   x  =  i0  sin  Bv 

hence,  substituting  r,  x,  Ov  i0,  gives 

A'  =  102. 
From  equation  (12), 


and,  substituting, 


sm02  -  0.527 


234  TRANSIENT  PHENOMENA 

approximately 

sin  02  =  0.527     and     62  =  32°; 

thus 

sin  62  =  0.527  £3'2°     =  0.558,     and    62  =  34°; 

thus 

sin  62  =  0.527  £3'4°     =  0.559,     and    02  =  34°. 

From  equations  (12)  and  (13)  it  follows : 

As      x  +  iQ  —  sin  (8  +  02)  =  i0  sin  02, 


Ae      x    '  -  i0  -  sin  (^  -  0J  =  i  sin  04; 

2 

eliminating  A  gives 

2  -  rt  sin  (5 


'T*  7*  I         /y» 

substituting  sin  8  =  -,  cos  5  =  -     —  *->  ^2  =  (r  +  rj2  +  x2,  and 

z  ,z 

substituting  for  r,  x,  6V  62,  gives  after  some  changes  : 

1.5-  1.04  r 


e-M,. 


1.1  -  r, 


calculating  by  approximation, 

assuming  rx  =  0.5, 

0.603  =  0.612; 

assuming  rs  =  0.51, 

0.597  -  0.602; 

assuming  r^  =  0.52, 

0.591  -  0.592; 
hence,  rl  =  0.52, 

and  z  =  2.124, 

5  =  70°. 


MECHANICAL  RECTIFICATION 


235 


Substituting  these  values  in  (12)  or  (13)  gives 

A  =-  113; 
hence,  as  final  equations,  we  have 

i  =  112  £-°-36'  +  34.6  sin  (6  -  70°), 
i'  =  102  £-°-10, 
i0  =  141.4  sin  0, 

and  it  =  i0  -  i', 

which  gives  the  following  results : 


Effec- 

Arith- 

Quantity. 

Instantaneous  Values. 

tive 

metic 

Value. 

Mean 

Value. 

e°  = 

30 

50 

70 

90 

110 

130 

146 

170 

190 

210 

i  = 
i'  = 

70.8 

70.5 

72.6 

74.9 

79.0 

80.4 

79.0 
79.0 

75.8 

73.2 

"7b'.8f 

75.2 

75.2 

?0  sin  0= 

70.8 

108 

133 

141.4 

133 

108 

79.0 

24.7 

-24.7 

-70.  8J 

100.0 

*i  = 

0 

37.5 

60.4 

66.5 

54.0 

27.6 

0 

(-51.1-48.5) 

0 

38.2 

'27^3 

(44.9) 

Curves  of  these  quantities  are  plotted  in  Fig.  55,  for  iQ  = 
100  sin  6. 

The  effective  value  of  the  rectified  current  is  75.2  amp.,  and 
this  current  is  fairly  constant,  pulsating  only  between  70.5  and 
80.4  amp.,  or  by  6.6  per  cent  from  the  mean;  that  is,  due  to  the 
self-inductance,  the  fluctuations  of  current  are  practically 
suppressed,  and  taken  up  by  the  non-inductive  shunt,  and  the 
arithmetic  mean  value  of  this  current  is  therefore  equal  to  its 
effective  value.  The  effective  value  of  the  shunt  current  is  38.2 
amp.,  and  this  current  is  unidirectional  also,  but  very  fluctuating. 
Its  arithmetic  mean  value  is  only  27.3  amp.;  that  is,  in  this 
circuit  a  continuous-current  ammeter  would  record  27.3,  an 
alternating  ammeter  38.2  amperes.  The  effective  value  of  the 
total  difference  between  alternating  and  rectified  current  (shunt 
plus  short-circuit  current)  is  44.9  amp. 

The  current  divides  between  the  inductive  rectified  circuit 
and  its  non-inductive  shunt,  not  in  proportion  to  their  respective 
impedances,  but  more  nearly,  though  not  quite,  in  proportion 


236 


TRANSIENT  PHENOMENA 


to  the  resistances;  that  is,  in  a  rectified  circuit,  self-inductance 
does  not  greatly  affect  the  intensity  of  the  current,  but  only  its 
character  as  regards  fluctuations. 


1W 
90 
80 
70 
60 

40 

£ 

10 
0 
-10 
-20 
-30 
-40 

s 

^ 

Si 

i  6 

^N, 

1  1 

/ 

N 

\ 

fc 

IOC 
0.5 
0.5 

si 
-5 

(- 

/ 

S 

\ 

t 

/ 

s 

v 

/ 

/ 

j 

\ 

/ 

/ 

x 

\^ 

"> 

s 

\ 

/ 

/ 

/ 

^ 

\ 

s 

/ 

/ 

/ 

'•il 

s 

w 

\ 

/ 

/ 

V 

? 

\ 

\ 

/ 

f 

/ 

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\ 

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\ 

^ 

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0         20        40        60         80        100      120       140       160      ISO       200 
Degrees  —  * 

Fig.  55.     Single-phase  current  rectification. 


2.   Single-phase  constant-potential  rectification. 

12.  Let  the  alternating  e.m.f.  eQ  sin  0  of  the  alternating  cir- 
cuit of  impedance  Z0  =  r0  —  jx0  .be  rectified  by  connecting  it 
at  the  moment  01  with  the  direct-current  receiver  circuit  of 
impedance  Z  =  r  —  jx  and  continuous  counter  e.m.f.  e,  dis- 
connecting it  therefrom  at  the  moment  TT  -  02,  and  closing 
during  the  time  from  n  —  02  to  n  +  Ol  the  alternating  circuit  by 
the  resistance  rv  the  direct-current  circuit  by  the  resistance  r2, 
then  connecting  the  circuits  again  in  series  in  opposite  direction, 
at  TT  +  6V  etc.,  as  shown  diagrammatically  by  Fig.  56,  where 

1 


rn  = 


1 


1 


rf  +r'"        r"  +  r"1 

1.   Then,  during  the  time  from  #t  to  n  —  02,  if  i1  =  current, 
the  differential  equation  is 

eQ  sin  d  -  e  -  i,  (r  +  r0)  -  (x  +  x0)  ^  =  0,  (1) 


MECHANICAL  RECTIFICATION 


which  is  integrated  by 

i\  =  A,  +  B 


,  sin  (0  - 


237 


(2) 


Fig.  56.     Single-phase  constant-potential  rectifying  commutator. 

Equation  (2)  substituted  in  (1)  gives 

e0  sin  0  -  e  -  (r  +  r0)  [A,  +  B^~^9  +  C,  sin  (6  -  9,)] 

-  (x  +x0)[-  a^Bf-*'  +  C,  cos  (0  -  dj]  =  0; 

or,  transposing, 

-  [  +  e  +  (r  +  r0)  AJ  +  Bfa*  [a,  (x  +  x0)  -  (r  +  r0)] 
+  sin  6  [e0  -  (r  +  r0)  Cl  cos  ^  -  (x  +  x0)  Ct  sin  ^J 
+  Ci  cos  0  [(r  +  r0)  sin  9t  -  (x  +  x0)  cos  dj  =  0; 

herefrom  it  follows  that 

e  +  (r  +rJAt  =  0, 
a,  (x  +  x0)  -  (r  +  r0)  =  0, 
eQ  -  (r  +  r0)  (71  cos  ^  -  (x  +  XQ)  C,  sin  ^  =  0, 

and 

(r  +  r0)  sin  9^  -  (x  +  x0)  cos  ^1  =  0; 


238 
hence 


TRANSIENT  PHENOMENA 


and 


tan  dt  = 


x+x( 

x+x, 
r  +rf 


and,  substituting  in  (2), 


(3) 


r  +r. 


r  + 


r+r0 


tan  dj  = 


2  +  (x 


(a: 


(4) 


2.   During  the  time  from  n  -  62  to  n  +  Ov  if  t2  =  current  in 
the  direct  circuit,  ia  =  current  in  alternating  circuit,  we  have 


Alternating-current  circuit: 


di 


-i3(r0  +  r,)  -  xQ-      =  0, 


(5) 


which  is  integrated  the  same  as  in  (1),  by 


_ r°  +  ^ 

Be        x° 


i   sn  0  -  x0  cos 


(r 


(6) 


MECHANICAL  RECTIFICATION  239 

Direct-current  circuit: 

-6-;2(r+r2)-z^  =  0,  (7) 

integrated  by 

_r_+_T20 

*'.  -  -  r-^72  +  B*s  ~  <8) 

At  ^  =  TT  —  <92,  however,  we  must  have 


and     i'2  at  0  =  n  +  6l  must  be  equal  to 

t\  at  #  =  0t,  and  opposite  to  i3  at  #  =  it  + 


(9) 


These  terminal  conditions  represent  four  equations,  which 
suffice  for  the  determination  of  the  three  remaining  integration 
constants,  Blf  B2,  Bs,  and  one  further  constant,  as  6l  or  62,  or 
rt  or  r2,  or  e;  that  is,  with  the  circuit  conditions  Z9,  Z,  rv  r2,  c0,  e 
chosen,  the  moment  ^  depends  on  i92  and  inversely. 

13.   Special  case: 

Z0  =  0,    r2  =  0,    e  =  0;  (10) 

that  is,  the  alternating  e.m.f.  eQ  sin  6  is  connected  to  the  circuit 
of  impedance  Z  =  r  -  jx  during  time  6^  to  n  -  62,  and  closed 
by  resistance  rv  while  the  rectified  circuit  is  short-circuited, 
during  time  TT  —  #2  to  n  +  Or 
The  equations  are  : 


1.   Time^  to  n  -  62: 


e 
+  -^  —  [r  sin  6  -  x  cos  6]. 

T     -j-  37 


2.   Time  TT  -  (9  to  n  + 


e0  sin  0 
io= 


(11) 


240  TRANS  I  EN  T  PHENOMENA 

The  terminal  conditions  now  assume  the  following  forms : 

At  0    =    7T    -    09f 


sin  S2  -f-  x  cos  02)  =  B2£ 
en 


at  0  =  TT  +  Ol  and  0X  respectively 


(12) 


These  four  equations  suffice  for  the  determination  of  the  two 
integration  constants  B1  and  B2,  and  two  of  the  three  rectifica- 
tion constants,  Ov  02J  rv  so  that  one  of  the  latter  may  be  chosen. 

Choosing  02,  the  moment  of  beginning  reversal,  the  equations 
(12)  transposed  and  expanded  give, 

sin  0 


sin  <92 
4-  x2 


(13) 
Bj_?>nVi<'-». 

^*i 

and 


-  B,  -  (r  sin  02  +  x  cos  0 


which  give  6lt  rv  B2,  Bl :  6l  is  calculated  by  approximation. 
Assuming,  as  an  example, 

e0  =  156  sin  6  (corresponding  to  110  volts  effective), 

Z  =  10-30f, 

and  ^ 

^  -  s  =  30°, 


MECHANICAL  RECTIFICATION 


241 


and 


by  equations  (13)  we  have : 

log  sin  01  =     -  0.3765  -  0.1448  0, 
Ol  -  21.7°, 
r,  -  7.63, 
B2  =  24.4, 
5,  -  12.8; 


and 
thus 


(15) 


and 

which  gives: 


=  12.8  e    ~3  +  1.56  (sin  6  -  3  cos  0), 

_  0 

=  24.4  r  3, 

=  20.5  sin  0, 


(16) 


0°, 

»V 

V 

V 

0°. 

V 

V 

V 

21  7 

7  55 

135 

10  27 

• 

30 
45 
60 

7.47 
7.7 
8.02 



150 
165 
180 

10.20 

10.2 
9.4 
8.6 

10.2 
5.3 
0 

75 
90 
105 

8.56 
9.18 
9  67 



195 
201.7 

7.9 
7.55 

-5.3 
-7.55 

120 

10  09 

The  mean  value  of  the  rectified  current  is  derived  herefrom 
as  8.92  amp.,  while  without  rectification  the  effective  value  of 

alternating    current    would    be  — .  =  3.48.      110  volts 


2v/§ 
effective  corresponds  to  -      -  110 


99  volts  mean,  which  in 


r  =  10  would  give  the  current  as  9.9  amp. 

Thus,  in  a  rectified  circuit,  self-inductance  has  little  effect 
besides  smoothing  out  the  fluctuations  of  current,  which  in  this 
case  varies  between  7.47  and  10.27,  with  8.92  as  mean,  while 
without  self-inductance  it  would  vary  between  0  and  15.6,  with 
9.9  as  mean,  and  without  rectification  the  current  would  be 
4.95  sin  (0  -  71.6°). 


242 


TRANSIENT  PHENOMENA 


As  seen,  in  this  case  the  exponential  or  transient  term  of 
current  largely  preponderates  over  the  permanent  or  sinusoidal 
term. 


40        60         80        100      : 
Degrees 


140       160       180 

Fig.  57.    Single-phase  e.m.f.  rectification. 


In  Fig.  57  is  shown  the  rectified  current  in  drawn  line,  the 
value  it  would  have  without  self-inductance,  and  the  value  the 
alternating  current  would  have,  in  dotted  lines. 


3.   Quarter-phase  constant-current  rectification. 

14.  In  the  quarter-phase  constant-current  arc  machine,  as 
the  Brush  machine,  two  e.m.fs.,  E1  =  e  cos  6  and  E2  =  e  sin  #, 
are  connected  to  a  rectifying  commutator,  so  that  while  the  first 
E^  is  in  circuit  E2  is  open-circuited.  At  the  moment  Ov  E2  is 
connected  in  parallel,  as  shown  diagrammatically  in  Fig.  58, 
with  Ev  and  the  rising  e.m.f.  in  E2  gradually  shifts  the  current 
i0  away  from  E1  into  E2,  until  at  the  moment  02,  E^  is  dis- 
connected and  E2  left  in  circuit. 

Assume  that,  due  to  the  superposition  of  a  number  of  such 
quarter-phase  e.m.fs.,  displaced  in  time-phase  from  each  other, 
and  rectified  by  a  corresponding  number  of  commutators  offset 
against  each  other,  and  due  to  self-inductance  in  the  external 
circuit,  the  rectified  current  is  practically  steady  and  has  the 
value  i0.  Thus  up  to  the  moment  6t  the  current  in  E1  is  i0,  in 


MECHANICAL  RECTIFICATION 


243 


E2  is  0.     From  61  to  62  the  current  in  E2  may  be  i;  thus  in  El  it 
is  i'2  =  i'0  -  t.    After  02,  the  current  in  El  is  0,  in  E2  it  is  i0. 

A  change  of  current  occurs  only  during  the  time  from  Ol  to  02, 
and  it  is  only  this  time  that  needs  to  be  considered. 


Fig.  58.     Quarter-phase  constant-current  rectifying  commutator. 

Let  Z  =  r  —  jx  =  impedance  per  phase,  where  x  =  2  njL ; 
then  at  the  time  t  and  the  corresponding  angle  0  =  2  njt  the 
difference  of  potential  in  El  is 

d  (i'0  -  t)      1 


6  cos  I?  —  (i0  —  i)  r  —  L 

di 
=  e  cos  0  -  (ia  -  i)  r  +  x  — ; 

the  difference  of  potential  in  E2  is 

di 
e  sin  0  -  ir  -  x  — ; 


(1) 


and,  since  these  two   potential  differences  are  connected    in 
parallel,  they  are  equal 


e  (sin  0  -  cos  6)  +  v  -2ir  -2x-    =  0.  (2) 

at/ 


244 


TRANSIENT  PHENOMENA 


The  differential  equation  (2)  is  integrated  by 

i  =  A  +  Bs-ae  +Ccos(0  -£); 


(3) 


thus 


-Csm(0  - 


and  substituting  in. (2), 

e  (sin  0  -  cos  0)  +  i0r  -  2  Ar  -  2  Bn'^  -2Cr  cos  (0  -  d) 
+  2  aBx£~ae  +2Cx  sin  (6  -  d)  =  Q- 

or,  transposed, 

(i0  -  2  A)  r  +  2  Bs~  °*(ax  -  r)  +  sin  d  [e  -  2  Cr  sin  8 

+  2Cx  cos  d]  -  cos  6  [e  +  2  Cr  cos  d  +  2  Cx  sin  d]  =  0; 

thus 

i0  -  2  A  =  0, 

ax  —  r  =  0, 

e  -  2  Cr  sin  d  +  2  Cx  cos  5  =  0, 
and  e  +  2  Cx  sin  d  +  2  Cr  cos  d  =  0, 

and  herefrom,  letting  -  =  tan  <r,  we  have 
r 

e  =  -  2Czsm(<r  -  d), 
e  =  -  2  (72  cos  (<r  -  d), 


r 
a  =  -; 


tan  r  = 


-  r 


x  +  r 


and 


C  =  - 


tan  (<r  -  §}  =  1, 

C  =— = 


(4) 


MECHANICAL  RECTIFICATION  245 

These  values  substituted  in  (3)  give 


t  -      +  Be~*    -  —  =  cos  (0  -  d), 

\/2  (r5  +  r2) 


tan  d  =  —  •  (5) 


At  0  =  0t,  i  =  0,  and  we  have 


0  =   «  +  Be    *1  -  —  -=  cos  (d,  -  d)] 

2  V2  (x2  +  r2) 


hence, 


Be~*  '  =     f  (d,  -  3)  -£;  (6) 

\/2  (z2  +  r2)  2 


substituting  in  (5),  we  have  the  equations  of  current  in  the  two 
coils  as  follows : 


e  (' 


\/2(za  + 
e 


cos  (#  — 
C^+r2) 

•(7) 

V2  (x2  +  r2)^  ' 

and 


+  -  -  cos  (0  -  5). 


246  TRANSIENT  PHENOMENA 

At  d  =  62,  i  =  i0;  thus 


r2) 

cos  (03  -  5)  =  0; 


x  2 

V2(x2  +  r2) 

by  e    x  1  an 
connecting  moments  dl  and  02,  as  follows: 


or,  multiplied  by  e    x  1  and  rearranged  ;  we  have  the  condition 


V2  (x2  +  r2)  ( 

=0 


and 

io 

2 


-       2  cos  (#2-£)    .  (8) 

Rearranged  equation  (8)  gives 


where  tan  d 


x  +r 


By  approximation,  from  this  equation  the  value  of  62,  corre- 
sponding to  a  given  Ov  is  derived. 
15.   Example : 

e  =  2000,  i0  =  10,  and  Z  =  10  -  40  j. 
Thus  d  =  31°  =  0.54  radians 


MECHANICAL  RECTIFICATION 


247 


and  i  =  5  +  [34.3  cos  (dl  -  31°)  -  5]  e~( 

-  34.3  cos  (0  -  31°), 
/  (02)  =  e0-25'2  [1  +  6.86  cos  (0a  -  31°)] 

=  £°'2501  [6.86  cos  (0,  -  31°)  -  1]. 

Substituting  for  Ov  30°  =   -t  ,  45°  =  j  ,  and  60°  =  ^  ,  respec- 

u  4  o 

tively,  gives: 


and 


,0       jr.. 

0  »  .*,  i  =  5  +  29.3e~   *V  "  *>  -  34.3  cos  (^-31°) 
6 

~~  -  31°) 


-£,  i=  5+  28.3 

£ ,  i  =  5  +  25 . 1 
3 


_34  .3  cos  (^- 


0i  = 

7T 

6 

0!  = 

7T 

4 

Oi  = 

IT 

3 

0° 

t 

ft 

i 

ii 

t 

ii 

€<! 

e\ 

25 

1810 

850 

30 
35 

0 

10 

1640 

1150 

40 
45 
50 
55 
60 

-0.9 
-0.6 
+  0  6 

10.9 
10.6 
9  4 

o' 

'".$' 

10 

"i!i 

o 

10 

1410 
1150 

1410 
1640 

65 

70 
75 
80 
85 
90 
95 
100 
105 
110 

3.0 
6^0 
9.9 
14^3 
19  1 

Yo 

4.0 
0.1 
-4.3 
—  9  1 

2.5 
5.1 

"s^e" 

12^8 
17^3 

7.5 
4^9 
1.4 
-2.8 
-7.3 

2^2 
5^4 
9.3 
13^8 

is  6 

7.8 
4.6 
'0.7 

-sis' 

-8  6 

850 
'520 
170 
-170 
-520 

1810 
1930 
1990 
1990 
1930 

115 

22.2 

-12.2 

-850 

1810 

These  values  are  plotted  in  Fig.  59,  together  with  el  and  er 
It  follows  then, 


90.2° 


88.6C 


91.7C 


248 


TRANSIENT  PHENOMENA 


The  actual  curves  of  an  arc  machine  differ,  however,  very 
greatly  from  those  of  Fig.  59.  In  the  arc  machine,  inherent  regu- 
lation for  constant  current  is  produced  by  opposing  a  very  high 
armature  reaction  to  the  field  excitation,  so  that  the  resultant 
m.m.f.,  or  m.m.f.  which  produces  the  effective  magnetic  flux,  is 


.40         5D 


90        100       110     120 


16 

y 

'^ 

12 

''/' 

10 

t2 

r  ' 

—  •— 

—  —  . 

--*^ 

'x/' 

\\ 

8   2000 

£ 

^ 

X 

<^ 

—  -I 

—  - 

•~~^ 

t*-"- 

—- 

h^ 

f 

^> 

—  - 

— 

4   1000 

^ 

.- 

•"J 

**** 

^ 

S. 

X 

* 

d2-2«50n 

^, 

_i- 

•"" 

~- 

^X 

.// 

1 

B    o 

•<30  f>      Q 

*""" 

ii 

.--- 

— 

^ 

x* 

-^ 

§ 

12 

-2  -500 

-  — 

—•  -^" 

<?~v 

-^ 

4  -1000 

3 

! 

! 

! 

s 

I 

^> 

"*•* 

^ 

-6 

N, 

•v\\ 

q 

i 

< 

q 

^ 

\ 

vv 
,\v 

10 

s^ 

t. 

-12 

\^ 

^'\V 

30        40        50         60        70        80         80        1QO      UO 
Degrees > 

Fig.  59.     Quarter-phase  rectification. 

small  compared  with  the  total  field  m.m.f.  and  the  armature 
reaction,  and  so  greatly  varies  with  a  small  variation  of  armature 
current.  As  result,  a  very  great  distortion  of  the  field  occurs, 
and  the  magnetic  flux  is  concentrated  at  the  pole  corner.  This 
gives  an  e.m.f.  wave  which  has  a  very  sharp  and  high  peak,  with 
very  long  flat  zero,  and  so  cannot  be  approximated  by  an  equiva- 
lent sine  wave,  but  the  actual  e.m.f.  curves  have  to  be  used  in  a 
more  exact  investigation. 


CHAPTER  IV. 

ARC   RECTIFICATION. 

I.  THE  ARC. 

16.  The  operation  of  the  arc  rectifier  is  based  on  the  charac- 
teristic of  the  electric  arc  to  be  a  good  conductor  in  one  direction 
but  a  non-conductor  in  the  opposite  direction,  and  so  to  permit 
only  unidirectional  currents. 

In  an  electric  arc  the  current  is  carried  across  the  gap  between 
the  terminals  by  a  bridge  of  conducting  vapor  consisting  of  the 
material  of  the  negative  or  the  cathode,  which  is  produced  and 
constantly  replenished  by  the  cathode  blast,  a  high  velocity 
blast  issuing  from  the  cathode  or  negative  terminal  towards  the 
anode  or  positive  terminal. 

An  electric  arc,  therefore,  cannot  spontaneously  establish 
itself.  Before  current  can  exist  as  an  arc  across  the  gap  between 
two  terminals,  the  arc  flame  or  vapor  bridge  must  exist,  i.e., 
energy  must  have  been  expended  in  establishing  this  vapor 
bridge.  This  can  be  done  by  bringing  the  terminals  into  contact 
and  so  starting  the  current,  and  then  by  gradually  withdrawing 
the  terminals  derive  the  energy  of  the  arc  flame  by  means  of  the 
current,  from  the  electric  circuit,  as  is  done  in  practically  all  arc 
lamps.  Or  by  increasing  the  voltage  across  the  gap  between  the 
terminals  so  high  that  the  electrostatic  stress  in  the  gap  repre- 
sents sufficient  energy  to  establish  a  path  for  the  current,  i.e.,  by 
jumping  an  electrostatic  spark  across  the  gap,  this  spark  is  fol- 
lowed by  the  arc  flame.  An  arc  can  also  be  established  between 
two  terminals  by  supplying  the  arc  flame  from  another  arc,  etc. 

The  arc  therefore  must  be  continuous  at  the  cathode,  but  may 
be  shifted  from  anode  to  anode.  Any  interruption  of  the  cathode 
blast  puts  out  the  arc  by  interrupting  the  supply  of  conducting 
vapor,  and  a  reversal  of  the  arc  stream  means  stopping  the 
cathode  blast  and  producing  a  reverse  cathode  blast,  which,  in 
general,  requires  a  voltage  higher  than  the  electrostatic  striking 

249 


250  TRANSIENT  PHENOMENA 

voltage  (at  arc  temperature)  between  the  electrodes.  With  an 
alternating  impressed  e.m.f.  the  arc  if  established  goes  out  at 
the  end  of  the  half  wave,  or  if  a  cathode  blast  is  maintained 
continuously  by  a  second  arc  (excited  by  direct  current  or 
overlapping  sufficiently  with  the  first  arc),  only  alternate  half 
waves  can  pass,  those  for  which  that  terminal  is  negative  from 
which  the  continuous  blast  issues.  The  arc,  with  an  alternating 
impressed  voltage,  therefore  rectifies,  and  the  voltage  range  of 
rectification  is  the  range  between  the  arc  voltage  and  the  electro- 
static spark  voltage  through  the  arc  vapor,  or  the  air  or  residual 
gas  which  may  be  mixed  with  it.  Hence  it  is  highest  with  the 
mercury  arc,  due  to  its  low  temperature. 

The  mercury  arc  is  therefore  almost  exclusively  used  for  arc 
rectification.  It  is  enclosed  in  an  evacuated  glass  vessel,  so  as 
to  avoid  escape  of  mercury  vapor  and  entrance  of  air  into  the 
arc  stream.  Due  to  the  low  temperature  of  the  boiling  point  of 
mercury,  enclosure  in  glass  is  feasible  with  the  mercury  arc. 

II.   MERCURY  ARC  RECTIFIER. 

17.  Depending  upon  the  character  of  the  alternating  supply, 
whether  a  source  of  constant  alternating  potential  or  constant 
alternating  current,  the  direct-current  circuit  receives  from  the 
rectifier  either  constant  potential  or  constant  current.  Depend- 
ing on  the  character  of  the  system,  thus  constant-potential 
rectifiers  and  constant-current  rectifiers  can  be  distinguished. 
They  differ  somewhat  from  each  other  in  their  construction  and 
that  of  the  auxiliary  apparatus,  since  the  constant-potential 
rectifier  operates  at  constant  voltage  but  varying  current,  while 
the  constant-current  rectifier  operates  at  varying  voltage.  The 
general  character  of  the  phenomenon  of  arc  rectification  is,  how- 
ever, the  same  in  either  case,  so  that  only  the  constant-current 
rectifier  will  be  considered  more  explicitly  in  the  following 
paragraphs. 

The  constant-current  mercury  arc  rectifier  system,  as  used 
for  the  operation  of  constant  direct-current  arc  circuits  from  an 
alternating  constant  potential  supply  of  any  frequency,  is  sketched 
diagrammatically  in  Fig.  60.  It  consists  of  a  constant-current 
transformer  with  a  tap  C  brought  out  from  the  middle  of  the 
secondary  coil  AB.  The  rectifier  tube  has  two  graphite  anodes 


ARC  RECTIFICATION 


251 


a,  6,  and  a  mercury  cathode  c,  and  usually  two  auxiliary  mercury 
anodes  near  the  cathode  c  (not  shown  in  diagram,  Fig.  60), 
which  are  used  for  excitation,  mainly  in  starting,  by  establishing 
between  the  cathode  c  and  the  two  auxiliary  mercury  anodes, 
from  a  small  low  voltage  constant-potential  transformer,  a  pair 
of  low  current  rectifying  arcs.  In  the  constant-potential  rectifier, 
generally  one  auxiliary  anode  only  is  used,  connected  through 
a  resistor  r  with  one  of  the  main  anodes,  and  the  constant- 


Fig.  60.    Constant-current 
mercury  arc  rectifier. 


Fig.  61.     Constant-potential 
mercury  arc  rectifier. 


current  transformer  is  replaced  by  a  constant-potential  trans- 
former or  compensator  (auto-transformer)  having  considerable 
inductance  between  the  two  half  coils  II  and  III,  as  shown  in 
Fig.  61.  Two  reactive  coils  are  inserted  between  the  outside 
terminals  of  the  transformer  and  rectifier  tube  respectively,  for 
the  purpose  of  producing  an  overlap  between  the  two  rectifying 
arcs,  ca  and  cb,  and  thereby  the  required  continuity  of  the  arc 
stream  at  c.  Or  instead  of  separate  reactances,  the  two  half  coils 
II  and  III  may  be  given  sufficient  reactance,  as  in  Fig.  61.  A 
reactive  coil  is  inserted  into  the  rectified  or  arc  circuit,  which 
connects  between  transformer  neutral  C  and  rectifier  neutral  c, 
for  the  purpose  of  reducing  the  fluctuation  of  the  rectified  current 
to  the  desired  amount. 

In  the  constant-potential  rectifier,  instead  of  the  transformer 
ACS  and  the  reactive  coils  A  a  and  Ba,  generally  a  compensator 
or  auto-transformer  is  used,  as  shown  in  Fig.  61,  in  which  the 


252  TRANSIENT  PHENOMENA 

two  halves  of  the  coil,  AC  and  BC,  are  made  of  considerable 
self-inductance  against  each  other,  as  by  their  location  on 
different  magnet  cores,  and  the  reactive  coil  at  c  frequently 
omitted.  The  modification  of  the  equations  resulting  herefrom  is 
obvious.  Such  auto-transformer  also  may  raise  or  lower  the 
impressed  voltage,  as  shown  in  Fig.  61. 

The  rectified  or  direct  voltage  of  the  constant-current  rectifier 
is  somewhat  less  than  one-half  of  the  alternating  voltage  supplied 
by  the  transformer  secondary  AB,  the  rectified  or  direct  current 
somewhat  more  than  double  the  effective  alternating  current 
supplied  by  the  transformer. 

In  the  constant-potential  rectifier,  in  which  the  currents  are 
larger,  and  so  a  far  smaller  angle  of  overlap  0  is  permissible,  the 
direct-current  voltage  therefore  is  very  nearly  the  mean  value 
of  half  the  alternating  voltage,  minus  the  arc  voltage,  which  is 
about  13  volts.  That  is,  if  e  =  effective  value  of  alternating 
voltage  between  rectifier  terminals  ab  of  compensator  (Fig.  61), 

2\/2 

hence  -    —  e  =  mean  value,  the  direct  current  voltage  is 

e0  =  — e.-  13. 

7T 

III.  MODE  OF  OPERATION. 

18.  Let,  in  Figs.  62  and  63,  the  impressed  voltage  between 
the  secondary  terminals  AB  of  an  alternating-current  trans- 
former be  shown  by  curve  I.  Let  C  be  the  middle  or  center  of 
the  transformer  secondary  AB.  The  voltages  from  C  to  A  and 
from  C  to  B  then  are  given  by  curves  II  and  III. 

If  now  A,B,C  are  connected  with  the  corresponding  rectifier 
terminals  a,  6,cand  at  c  a  cathode  blast  maintained,  those  currents 
will  exist  for  which  c  is  negative  or  cathode,  i.e.,  the  current 
through  the  rectifier  from  a  to  c  and  from  b  to  c,  under  the 
impressed  e.m.fs.  II  and  III,  are  given  by  curves  IV  and  V,  and 
the  current  derived  from  c  is  the  sum  of  IV  and  V,  as  shown  in 
curve  VI. 

Such  a  rectifier  as  shown  diagrammatically  in  Fig.  62  requires 
some  outside  means  for  maintaining  the  cathode  blast  at  c,  since 
the  current  in  the  half  wave  1  in  curve  VI  goes  down  to  zero  at 


ARC  RECTIFICATION 


253 


the  zero  value  of  e.m.f.  Ill  before  the  current  of  the  next  half 
wave  2  starts  by  the  e.m.f.  II. 

It  is  therefore  necessary  to  maintain  the  current  of  the  half 
wave  1  beyond  the  zero  value  of  its  propel- 
ling impressed  e.m.f.  Ill  until  the  current  of 
the  next  half  wave  2  has  started,  i.e.,  to 
overlap  the  currents  of  the  successive  half 
waves.  This  is  done  by  inserting  reactances 
into  the  leads  from  the  transformer  to  the 
rectifier,  i.e.,  between  A  and  a,  B  and  b  respec- 
tively, as  shown  in  Fig.  60.  The  effect  of 
this  reactance  is  that  the  current  of  half  wave 
1,  V,  continues  beyond  the  zero  of  its  im- 
pressed e.m.f.  Ill  i.e.,  until  the  e.m.f.  Ill  has 
died  out  and  reversed,  and  the  current  of  the 
half  wave  2,  IV,  started  by  e.m.f.  II;  that  is, 
the  two  half  waves  of  the  current  overlap, 
and  each  half  wave  lasts  for  more  than  half 
a  period  or  180  degrees. 

The  current  waves  then  are  shown  in  curve 
VII.    The  current  half  wave  1  starts  at  the  zero  value  of  its 
e.m.f.  Ill,  but  rises  more  slowly  than  it  would  without  react- 


Fig.  62.  Constant- 
current  mercury 
arc  rectifier. 


I 

/ 

\ 

/ 

\ 

^~. 

s 

^- 

1 

\ 

/ 

\ 

s^ 

\ 

1 

^ 

J 

\ 

2 

f 

1 

/ 

*S 

^ 

*-* 

\^ 

-v 

s— 

(s, 

j 

/ 

/ 

11 
III 

1 

/\ 

\\ 

^ 

\  — 

s 

v 

•^ 

/" 

\ 

/ 

\ 

/ 

/ 

/ 

\ 

/ 

^ 

\ 

\£ 

*-> 

k 

/ 

<r 

\ 

^< 

y 

VI 

vn 

/ 

1 

\ 

/ 

v^ 

V 

\ 

\ 

4p 

- 

^ 

s 

T" 

; 

^ 

v 

s 

— 

s 

/ 

/ 

!/ 

/ 

i 

X 

i 

x 

2 

X 

X 

r^ 

-s 

S-^1 

c^ 

— 

L- 

xn 


XIII 


Fig.  63.     E.m.f.  and  current  waves  of  constant-current  mercury  arc  rectifier. 

ance,  following  essentially  the  exponential  curve  of  a  starting 
current  wave,  and  the  energy  which  is  thus  consumed  by  the 
reactance  as  counter  e.m.f.  is  returned  by  maintaining  the 


254  TRANSIENT  PHENOMENA 

current  half  wave  1  beyond  the  e.m.f.  wave,  i.e.,  beyond  180 
degrees,  by  00  time-degrees,  so  that  it  overlaps  the  next  half 
wave  2  by  00  time-degrees. 

Hereby  the  rectifier  becomes  self-exciting,  i.e.,  each  half  wave 
of  current,  by  overlapping  with  the  next,  maintains  the  cathode 
blast  until  the  next  half  wave  is  started. 

The  successive  current  half  waves  added  give  the  rectified  or 
unidirectional  current  curve  VIII. 

During  a  certain  period  of  time  in  each  half  wave  from  the  zero 
value  of  e.m.f.  both  arcs  ca  and  cb  exist.  During  the  existence 
of  both  arcs  there  can  be  no  potential  difference  between  the 
rectifier  terminals  a  and  b,  and  the  impressed  e.m.f.  between  the 
rectifier  terminals  a  and  b  therefore  has  the  form  shown  in  curve 
IX,  Fig.  63,  i.e.,  remains  zero  for  #0  time-degrees,  and  then  with 
the  breaking  of  the  arc  of  the  preceding  half  wave  jumps  up  to 
its  normal  value. 

The  generated  e.m.f.  of  the  transformer  secondary,  however, 
must  more  or  less  completely  follow  the  primary  impressed  e.m.f. 
wave,  that  is,  has  a  shape  as  shown  in  curve  I,  and  the  difference 
between  IX  and  I  must  be  taken  up  by  the  reactance.  That  is, 
during  the  time  when  both  arcs  exist  in  the  rectifier,  the  a.  c. 
reactive  coils  consume  the  generated  e.m.f.  of  the  transformer 
secondary,  and  the  voltage  across  these  reactive  coils,  therefore, 
is  as  shown  in  curve  X.  That  is,  the  reactive  coil  consumes 
voltage  at  the  start  of  the  current  of  each  half  wave,  at  x  in 
curve  X,  and  produces  voltage  near  the  end  of  the  current,  at  y. 
Between  these  times,  the  reactive  coil  has  practically  no  effect  and 
its  voltage  is  low,  corresponding  to  the  variation  of  the  rectified 
alternating  current,  as  shown  in  curve  XI.  That  is,  during  this 
intermediary  time  the  alternating  reactive  coils  merely  assist  the 
direct-current  reactive  coil. 

Since  the  voltage  at  the  alternating  terminals  of  the  rectifier, 
a,  6,  has  two  periods  of  zero  value  during  each  cycle,  the  rectified 
voltage  between  c  and  C  must  also  have  the  same  zero  periods, 
and  is  indeed  the  same  curve  as  IX,  but  reversed,  as  shown  in 
curve  XII. 

Such  an  e.m.f.  wave  cannot  satisfactorily  operate  arcs,  since 
during  the  zero  period  of  voltage  XII  the  arcs  go  out.  The 
voltage  on  the  direct-current  line  must  never  fall  below  the 
"counter  e.m.f."  of  the  arcs,  and  since  the  resistance  of  this 


ARC  RECTIFICATION  255 

circuit  is  low,  frequently  less  than  10  per  cent,  it  follows  that  the 
total  variation  of  direct-current  line  voltage  must  be  below 
10  per  cent,  i.e.,  the  voltage  practically  constant,  as  shown  by  the 
straight  line  in  curve  XII.  Hence  a  high  reactance  is  inserted 
into  the  direct-current  circuit,  which  consumes  the  excess  voltage 
during  that  part  of  curve  XII  where  the  rectified  voltage  is 
above  line  voltage,  and  supplies  the  line  voltage  during  the 
period  of  zero  rectified  voltage.  The  voltage  across  this  reactive 
coil,  therefore,  is  as  shown  by  curve  XIII. 

IV.  CONSTANT-CURRENT  RECTIFIER. 

19.  The  angle  of  overlap  00  of  the  two  arcs  is  determined  by 
the  desired  stability  of  the  system.  By  the  angle  00  and  the 
impressed  e.m.f.  is  determined  the  sum  total  of  e.m.fs.  which 
has  to  be  consumed  and  returned  by  the  a.  c.  reactive  coil,  and 
herefrom  the  size  of  the  a.  c.  reactive  coil. 

From  the  angle  00  also  follows  the  wave  shape  of  the  rectified 
voltage,  and  therefrom  the  sum  total  of  e.m.f.  which  has  to  be 
given  by  the  d.  c.  reactive  coil,  and  hereby  the  size  of  the  d.  c. 
reactive  coil  required  to  maintain  the  d.  c.  current  fluctuation 
within  certain  given  limits. 

The  efficiency,  power  factor,  regulation,  etc.,  of  such  a  mercury 
arc  rectifier  system  are  essentially  those  of  the  constant-current 
transformer  feeding- the  rectifier  tube. 

Let  /  =  frequency  of  the  alternating-current  supply  system, 
i0  =  mean  value  of  the  rectified  direct  current,  and  a  =  the  pulsa- 
tion of  the  rectified  current  from  the  mean  value,  i.e.,  i0  (1  +  a) 
the  maximum  and  i0  (1  -  a)  the  minimum  value  of  direct  cur- 
rent. A  pulsation  from  a  mean  of  20  to  25  per  cent  is  permissible 
in  an  arc  circuit.  The  total  variation  of  the  rectified  current 
then  is  2  aiOJ  i.e.,  the  alternating  component  of  the  direct  current 

has  the  maximum  value  ai0,  hence  the  effective  value  — i_  i0    (or 

for  a  =  0.2,  0.141 10)  and  the  frequency  2/.  Hysteresis  and  eddy 
losses  in  the  direct-current  reactive  coil,  therefore,  correspond 
to  an  alternating  current  of  frequency  2f  and  effective  value 

a 
-——  i0,  or  about  0.141  -iQ,  i.e.,  are  small  even  at  relatively  high 

densities. 


256 


TRANSIENT  PHENOMENA 


In  the  alternating-current  reactive  coils  the  current  varies, 
unidirectionally,  between  0  and  i0  (1  +  a),  i.  e.,  its  alternating 

i0  and   the    effec- 


component  has  the  maximum  value 

tive  value-— =  iQ   (or,  for  a  =  +  0.2,  0.425  i0)    and  the  fre- 


JU 


quency/.     The   hysteresis   loss,   therefore,  corresponds   to   an 
alternating  current  of  frequency  /  and  effective  value  %> 

or  about  0.425  i0. 

With  decreasing  load,  at  constant  alternating-current  supply, 
the  rectified  direct  current  slightly  increases,  due  to  the  increas- 
ing overlap  of  the  rectifying  arcs,  and  to  give  constant  direct 
current  the  transformer  must  therefore  be  adjusted  so  as  to 
regulate  for  a  slight  decrease  of  alternating-current  output  with 
decrease  of  load. 


V.  THEORY  AND  CALCULATION. 

20.   In  the  constant-current  mercury-arc  rectifier  shown  dia- 
grammatically  in  Fig.  64,  let  e  sirr  6  =  sine  wave  of  e.m.f.  im- 
pressed between  neutral  and  outside  of 
alternating-current   supply  to  the  rec- 
tifier; that  is,  2  e  sin  6  =  total  secondary 
generated  e.m.f.  of  the  constant-current 
transformer;   Z1  =  r1  --  jx1  =   imped- 
ance of  the  reactive  coil  in  each  anode 
circuit  of    the    rectifier    ("  alternating- 
current  reactive  coil")?  inclusive  of  the 
internal  self-inductive   impedance    be- 
tween the  two  halves  of  the  transformer 
secondary  coil;  t\  and  i2  =  anode  cur- 
rents,  counted  in  the  direction    from 
anode  to  cathode;  ea  =  counter  e.m.f. 
Fig.  64.    Constant-current     of  rectifying  arc,  which  is  constant;  Z0  = 
mercury  arc  rectifier.         r0  —  jx0  =  impedance  of    reactive    coil 
in  rectified  circuit  ("  direct-current  re- 
active coil");  Z2  =  r2  ~  JX2  =  impedance  of  load  or  arc-lamp 
circuit;  e/  =  counter  e.m.f.  in  rectified  circuit,  which  is  con- 


ARC  RECTIFICATION  257 

stant  (equal  to  the  sum  of  the  counter  e.m.fs.  of  the  arcs  in  the 
lamp  circuit)  ;  #0  =  angle  of  overlap  of  the  two  rectifying  arcs, 
or  overlap  of  the  currents  it  and  i2  ;  i0  =  rectified  current  during 
the  period,  0  <  0  <  00,  where  both  rectifying  arcs  exist,  and  i0'  = 
rectified  current  during  the  period,  00  <  6  <  n,  where  only  one 
arc  or  one  anode  current  i1  exists. 

Let  e0  =  eQ'  +  ea.  =  total  counter  e.m.f.  in  the  rectified  cir- 
cuit and  Z  =  r  —  jx  =  (rt  +  r0  +  r2)  -  j  (xl  +  x0  -f  x2)  =  total 
impedance  per  circuit;  then  we  have 

(a)  During  the  period  when  both  rectifying  arcs  exist, 

o  <  e  <  00, 

%  =  h  +  v  C1) 

In  the  circuit  between  the  e.m.f.  2  e  sin  d,  the  rectifier  tube, 
and  the  currents  \  and  iv  according  to  Kirchhoff's  law,  it  is, 
Fig.  64, 

di  di 

2esin0  -  v\  -^-+^2+^1-      =  0-  (2) 


In  the  circuit  from  the  transformer  neutral  over  e.m.f.  e  sin  6, 
current  iv  rectifier  arc  ea  and  rectified  circuit  i'0,  back  to  the 
transformer  neutral,  we  have 

di~  .  di0         .  di0 

esmO-r^-x^  -ea  -  r,i0  -  x0-^-  r2i0  -  x2—  -  e,'  - 

or, 

e  sin  d  -  rlil  -  x^  -  (r0  +  ra)  i0  -  (x0  +  x2)  -^  -  e0  =  0.     (3) 

(6)  During  the  period  when  only  one  rectifying  arc  exists, 

00  <  e  <  x, 
h  =  V; 

hence,  in  this  circuit, 

di  '  di  ' 

e  sin  0  -vY  -  x^  -  (r0  +  r2)  iQ'  -  (x0  +  x2)  -±  -e.  =  0.    (4) 


258 


TRANSIENT  PHENOMENA 


Substituting  (1)  in  (2)  and  combining  the  result  (5)  of  this 
substitution  with  (3)  gives  the  differential  equations  of  the  rec- 
tifier: 

2  e  sin  0  +  r,  (i0  -  2  i,}  +  X^'fa  -  2  ij  -  0,  (5) 


and 


2  e0  +  (2  r  -  rj  i0  +  (2x-  xj     °  -  0, 

di  ' 
e  sin  6  —  e    —  n''  —  x  —-  =  0. 


(6) 

(7) 


In  these  equations,  iQ  and  ^  apply  for  the  time,  0  <  0  <  00, 
ij  for  the  time,  00  <  d  <  n. 

21.   These  differential  equations  are  integrated  by  the  func- 
tions 

in  -2i,  =  Ae~a9  +  A'  sin  (6  -  /?),  (8) 

L=B*-»  +B',  (9) 


and 


(10) 


Substituting  (8),  (9),  and  (10)  into  (5),  (6),  and  (7)  gives 
three  identities  : 

2  e  sin.fl+A7  [r,  sin  (0-0)  +x,  cos  (6-p)}+Ae-ae  (r^-ax^)  =0. 

2  e0+B'(2  r-rj  +5e-M  [(2  r-rj  -b  (2  x-xj]  =  0, 
and 
esmd-e0-C"[rsm(d-r)+xcos(0-r)]-C'r-Ce-c9(r-cx)=0', 


hence, 


and 


r,  -  ax,  -  0, 

(2  r  -  rt)  -  b  (2  x  -  x,)  =  0, 
r  —  ex  =  0. 

e0  +  C'r  =  0, 

2  e  +  A'  (r1  cos  /?  +  x1  sin  /?)  =  0, 
A7  (rt  sin  /?  -  xl  cos  /?)  =  0, 
e  —  C"  (r  cos  7-  +  £  sin  ?-)  =  0, 
C"  (r  sin  ^  -  x  cos  ?)  =  0. 


(11) 


ARC  RECTIFICATION 


259 


Writing 


?    =  v/r  2   4-  r  2 
z\  ~        ri     r  *t.  i 

tan  <*,=—, 


and 


tan  a:  =  - 
r 


(12) 


(13) 


Substituting  (12)  and  (13)  gives  by  resolving  the  9  equations 
(11)  the  values  of  the  coefficients  a,  b,  c,  A',  J5',  C",  C",  ft  r: 


(14) 


r 

c  =  -> 
x 


r—  a, 


2e 


B'-          2e« 


2r-r1 


C"  =  +-, 

z 


and  thus  the  integral  equations  of  the  rectifier  are 


and 


io-Zi,  =A£~ae  -     -sin  (0  -a,), 


2r  -  r. 


r      2 


(15) 


(16) 


(17) 

(18) 
(19) 
(20) 


260 


TRANSIENT  PHENOMENA 


where  a,  b,  c  are  given  by  equations  (14),  a  and  al  by  equations 
(12)  and  (13),  and  A,  J3,  C  are  integration  constants  given  by  the 
terminal  conditions  of  the  problem. 
22.   These  terminal  conditions  are : 


=  o, 


0     f« 


and 


(21) 


That  is,  at  0  =  0  the  anode  current  il  =  0.  After  half  a 
period,  or  n  =  180°,  the  rectified  current  repeats  the  same 
value.  At  6  =  00,  all  three  currents  iv  iw  iQ'  are  identical. 

The  four  equations  (21)  determine  four  constants,  A,  B,  C,  00. 

Substituting  these  constants  in  equations  (18),  (19),  (20) 
gives  the  equations  of  the  rectified  current  i0,  iQ',  and  of  the 
anode  currents  i^  and  i2  =  i0  —  iv  determined  by  the  constants 
of  the  system,  Z,  Zv  e0,  and  by  the  impressed  e.m.f.,  e. 

In  the  constant-current  mercury-arc  rectifier  system  of  arc 
lighting,  e,  the  secondary  generated  voltage  of  the  constant- 
current  transformer,  varies  with  the  load,  by  the  regulation  of 
the  transformer,  and  the  rectified  current,  iQ,  i0',  is  required  to 
remain  constant,  or  rather  its  average  value. 

Let  then  be  given  as  condition  of  the  problem  the  average 
value  i  of  the  rectified  current,  4  amperes  in  a  magnetite  or 
mercury  arc  lamp  circuit,  5  or  6.6  or  9.6  amperes  in  a  carbon 
arc  lamp  circuit. 

Assume  as  fair  approximation  that  the  pulsating  rectified 
current  i0,  iQr  has  its  mean  value  i  at  the  moment,  6  =  0.  This 
then  gives  the  additional  equation 

Kol.-o  -  *>  (22) 

and  from  the  five  equations  (21)  and  (22)  the  five  constants 
A,  B,  (7,  00,  e  are  determined. 

Substituting  (22),  (18),  (19),  (20)  inequations  (21)  gives 


A  •  ' 

A  =  i  ---  sin 


a 


C  =S*i  +   -°-sma 
r       z 


(23) 


ARC  RECTIFICATION  261 


_          sn    a_ 


zl  2  r  - 


(24) 


Substituting  (23)  in  (24)  gives 

2  e(  ) 

•  j  €    '  9°  sin  al  -  sin  (al  —  #0)  >  =  i 

z  1  \  ) 


2r  - 

and 


-«-*•         (25) 

^  r  —  TJ  f  ) 

w~*o)  sin  «  +  sin  (a  —  60] 


2  r  -  rt  (  )      r    < 

and  eliminating  e  from  these  two  equations  gives 


)     t(2r-r,){  \ 

(27) 

Equation  (27)  determines  angle  00,  and  by  successive  substitu- 
tion in  (26),  (23),  e,  A,  B,  C  are  found. 

Equation  (27)  is  transcendental,  and  therefore  has  to  be  solved 
by  approximation,  which  however  is  very  rapid. 

As  first  approximation,  a  =  6  =  c  =  0;  a=a1  =  90°  or  - 

2i 

and  substituting  these  values  in  (27)  gives 


cosfl,      2* 


V         W  +  W 


1  -  cos  dl       zl 


262  TRANSIENT  PHENOMENA 

and 


(28) 


This  value  of  Ol  substituted  in  the  exponential  terms  of  equa- 
tion (27)  gives  a  simple  trigonometric  equation  in  00,  from  which 
follows  the  second  approximation  02,  and,  by  interpolation,  the 
final  value, 


23.   For  instance,  let  e0  =  950,  i  =  3.8,  the  constants  of  the 
circuit  being  Zl  =  10  -  185  /  and  Z  =  50  -  1000  /. 

Herefrom  follows 

a  =  0.054,  b  =  0.050,  and  c  =  0.050,  (14) 

«A  =  86.9°  and  a  =  87.1°.  (15) 

From  equation  (28)  follows  as  first  approximation,  #t  =  47.8°; 
as  second  approximation,  #2  =  44.2°. 
Hence,  by  (29), 

0  =  44.4°. 

Substituting  a  in  (26)  gives  e  =  2100, 
hence,  the  effective  value  of  transformer  secondary  voltage, 

2  e 

J—  =  2980  volts 

V2 

and,  from  (23), 

A  =  --  18.94,  B  =  24.90,  C  =  24.20. 
Therefore,  the  equations  of  the  currents  are 
i0  =  24.90s-0-0500-  21.10, 

V=  24.20  e-0'050*  -  19.00  +2.11  sin  (6  -  87.1°), 
i,  =  12.45  e-0'050*  +  9.47  £-°-0540-  10.58  + 11.35  sin  (0  -  86.9°), 

and 


ARC  RECTIFICATION 


263 


The  effective  or  equivalent  alternating  secondary  current  of 
the  transformer,  which  corresponds  to  the  primary  load  current, 
that  is,  primary  current  minus  exciting  current,  is 


V  V-t  Vt 


From  these  equations  are  calculated  the  numerical  values  of 
rectified  current  iot  t'0',  of  anode  current  iv  and  of  alternating 
current  i' ,  and  plotted  as  curves  in  Fig.  65. 


\ 


\ 


Fig.  65.    Current  waves  of  constant-current  mercury  arc  rectifier. 


24.  As  illustrations  of  the  above  phenomena  are  shown  in 
Fig.  66  the  performance  curves  of  a  small  constant-current  rec- 
tifier, and  in  Figs.  67  to  76  oscillograms  of  this  rectifier. 

Interesting  to  note  is  the  high  frequency  oscillation  at  the  ter- 
mination of  the  jump  of  the  potential  difference  cC  (Fig.  60) 
which  represents  the  transient  term  resulting  from  the  electro- 
static capacity  of  the  transformer.  At  the  end  of  the  period  of 
overlap  of  the  two  rectifying  arcs  one  of  the  anode  currents  reaches 


264 


TRANSIENT  PHENOMENA 


=£ 


24 


20 


GO     |   12 

o 
40      n     8 


100     200     300      400     500     600     700     800     900    1000    1100 
Volt  Load 

Fig.  66.     Results  from  tests  made  on  a  constant-current  mercury  arc  rectifier. 


\ 


Fig.  67.     Supply  e.m.f.  to  constant-current  rectifier. 

A    A    A    , 

/   V    V    V 

Fig.  68.     Secondary  terminal  e.in.f.  of  transformer. 


Fig.  69.     E.m.f.  across  a.c.  reactive  coils. 


r\ 


.  , 

i/ 

Fig.  70.     Alternating  e.m.f.  impressed  upon  rectifier  tube. 


ARC  RECTIFICATION 


265 


w\r\r\r\ 


Fig.  71.     Unidirectional  e.m.f.  produced  between  rectifier  neutral  and 
transformer  neutral. 


V 


Fig.  72.     E.m.f.  across  d.c.  reactive  coils. 


Fig.  73.     Rectified  e.m.f.  supplied  to  arc  circuit. 


Fig.  74.    Primary  supply  current. 


Fig.  75.     Current  in  rectifying  arcs. 


Fig.  76.     Rectified  current  in  arc  circuit. 


266  TRANSIENT  PHENOMENA 

zero  and  stops,  and  so  its  L  —  abruptly  changes;  that  is,  asud- 

ctt 

den  change  of  voltage  takes  place  in  the  circuit  aACDc  or 
bBCDc.  Since  this  'cuit  contains  distributed  capacity,  that 
of  the  transformer  C'  ^BC  respectively,  the  line,  etc.,  and 
inductance,  an  oscillatk  ^.-.esults  of  a  frequency  depending  upon 
the  capacity  and  inductance,  usually  a  few  thousand  cycles  per 
second,  and  of  a  voltage  depending  upon  the  impressed  e.m.f.; 

that  is,  the  L  —  of  the  circuit.     An  increase  of  inductance  L 
dt 

di 
increases  the  angle  of  overlap  and  so  decreases  the  — ,  hence  does 

CLL 

not  greatly  affect  the  amplitude,  but  decreases  the  frequency  of 

this  oscillation.     An  increase  of  —  at  constant  L,  as  resulting 

dt 

from  a  decrease  of  the  angle  of  overlap  by  delayed  starting  of 
the  arc,  caused  by  a  defective  rectifier,  however  increases  the 
amplitude  of  this  oscillation,  and  if  the  electrostatic  capacity  is 
high,  and  therefore  the  damping  out  of  the  oscillation  slow,  the 


Fig.  77.    E.m.f.  between  rectifier  anodes. 

oscillation  may  reach  considerable  values,  as  shown  in  oscillo- 
gram,  Fig.  77,  of  the  potential  difference  db.  In  such  cases,  if 
the  second  half  wave  of  the  oscillation  reaches  below  the  zero 
value  of  the  e.m.f.  wave  db,  the  rectifying  arc  is  blown  out  and 
a  disruptive  discharge  may  result. 


ARC  RECTIFICATION  267 


VI.  EQUIVALENT  SINE  WAVES. 

25.  The  curves  of  voltage  and  current  in  the  mercury-arc 
rectifier  system,  as  calculated  in  the  p!neding  from  the  con- 
stants of  the  circuit,  consist  of  succe,  ,  Sections  of  exponential 


or  of  exponential  and  trigonometric  c 

In  general,  such  wave  structures,  built  up  of  successive  sections 
of  different  character,  are  less  suited  for  further  calculation. 
For  most  purposes,  they  can  be  replaced  by  their  equivalent 
sine  waves,  that  is,  sine  waves  of  equal  effective  value  and  equal 
power. 

The  actual  current  and  e.m.f.  waves  of  the  arc  rectifier  thus 
may  be  replaced  by  their  equivalent  sine  waves,  for  general 
calculation,  except  when  investigating  the  phenomena  resulting 
from  the  discontinuity  in  the  change  of  current,  as  the  high 
frequency  oscillation  at  the  end  and  to  a  lesser  extent  at  the 
beginning  of  the  period  of  overlap  of  the  rectifying  arcs,  and 
similar  phenomena. 

In  a  constant-current  mercury  arc  rectifier  system,  of  which 
the  exact  equations  or  rather  groups  of  equations  of  currents 
and  of  e.m.fs.  were  given  in  the  preceding,  let  %  =  the  mean 
value  of  direct  current;  eQ  =  the  mean  value  of  direct  or  rectified 
voltage;  i  =  the  effective  value  of  equivalent  sine  wave  of 
secondary  current  of  transformer  feeding  the  rectifier;  e  =  the 
effective  value  of  equivalent  sine  wave  of  total  e.m.f.  generated 

^> 

in  the  transformer  secondary  coils,   hence,  -  =  the  effective 

ft 

equivalent  sine  wave  of  generated  e.m.f.  per  secondary  trans- 
former coil,  and  #0  =  the  angle  of  overlap  of  rectifying  arcs. 

The  secondary  generated  e.m.f.,  e,  is  then  represented  by  a 
sine  wave  curve  I,  Fig.  78,  with  e  \/2  as  maximum  value. 

Neglecting  the  impedance  voltage  of  the  secondary  circuit 
during  the  time  when  only  one  arc  exists  and  the  current  changes 
are  very  gradual,  the  terminal  voltage  between  the  rectifier 
anodes,  ev  is  given  by  curve  II,  Fig.  78,  with  e  V2  as  maximum 
value.  This  curve  is  identical  with  e,  except  during  the  angle 
of  overlap  #0,  when  e^  is  zero.  Due  to  the  impedance  of  the) 
reactive  coils  in  the  anode  leads,  curve  II  differs  slightly  from  I, 
but  the  difference  is  so  small  that  it  can  be  neglected  in  deriving 


268 


TRANSIENT  PHENOMENA 


the  equivalent  sine  wave,  and  this  impedance  considered  after- 
wards as  inserted  into  the  equivalent  sine-wave  circuit. 
The  rectified  voltage,  e2,  is  then  given  by  curve  III,  Fig.  78, 

e    /—        e 
with  a  maximum  value  of  -  v  2  =  —/=-  and  zero  value  during 

the  angle  of  overlap  00,  or  rather  a  value  =  ea,  the  e.m.f.  con- 
sumed by  the  rectifying  arc  (13  to  18  volts). 


ii 


in 


IV 


VI 


VII 


viii 


Fig.  78.     E.m.f.  and  current  curves  in  a  mercury  arc  rectifier  system. 

The  direct  voltage  e0,  when  neglecting  the  effective  resistance 
of  the  reactive  coils,  is  then  the  mean  value  of  the  rectified 
voltage,  e2,  of  curve  III,  hence  is 


en  = 


elf. 

=  — :  -    /    SI 

v/2  *V* 


sin  d  dd 


ARC  RECTIFICATION 

_  e  (1  +  cos  00)  m 


y>     — —     0 
fy      C/. 


269 


If  ea  =  the  mercury  arc  voltage,  r0  =  the  effective  resistance  of 
reactive  coils  and  i0  =  the  direct  current,  more  correctly  it  is 


The  effective  alternating  voltage  between  the  rectifier  anodes 
is  the  Vmean  square  of  ev  curve  II,  hence  is 


2  00  -  sin  2  0 

27T 


and  the  drop  of  voltage  in  the  reactive  coils  in  the  anode  leads, 
caused  by  the  overlap  of  the  arcs,  thus  is 


-  sin  2  0( 

2x 


26.  Let  iv  =  the  maximum  variation  of  direct  current  from 
mean  value  i'0,  hence,  i2  =  i0  +  if  ='  the  maximum  value  of 
rectified  current,  and  therefore  also  the  maximum  value  of 
anode  current. 

The  anode  current  thus  has  a  maximum  value  iv  and  each 
half  wave  has  a  duration  TT  +  00,  as  shown  by  curve  IV,  Fig.  78. 

The  direct  current,  i0,  is  then  given  by  the  superposition  or 
addition  of  the  two  anode  currents  shown  in  curves  V,  and  is 
given  in  curve  VI. 


270  TRANSIENT  PHENOMENA 

The  effective  value  of  the  equivalent  alternating  secondary 
current  of  the  transformer  is  derived  by  the  subtraction  of  the 
two  anode  currents,  or  their  superposition  in  reverse  direction, 
as  shown  by  curves  VII,  and  is  given  by  curve  VIII. 

Each  impulse  of  anode  current  covers  an  angle  n  +  00,  or 
somewhat  more  than  one  half  wave. 

Denoting,  however,  each  anode  wave  by  n,  that  is,  considering 
each  anode  impulse  as  one  half  wave  (which  corresponds  to  a 

Ifower  frequency—       -),   then,  referred  to  the  anode  impulse 

x  +  V 

as  half  wave,  the  angle  of  overlap  is 

TT 

0,    = 


1        ^     , 

71    -j- 

The  direct  current,  i0,  is  the  mean  value  of  the  anode  current 
curves  V,  VI,  and,  assuming  the  latter  as  equivalent  sine  waves 
of  maximum  value  i2  =  i0  +  i' ',  the  direct  current,  i0,  is 

sin  d'  dd' 


u1  i/o 

2L 


-  0, 

2  (n  +  00)  L 


and 


or, 


and  the  pulsation  of  the  direct  current,  if  =  i2  —  i0,  is 

7T2  ) 


The  effective  value  of  the  secondary  current,  as  equivalent 
sine  wave  in  one  transformer  coil,  is  the  \/  mean  squiare  of  curves 
VII,  VIII,  or,  assuming  this  current  as  existing  in  both  trans- 
former secondary  coils  in  series  —  actually  it  alternates,  one  half 


ARC  RECTIFICATION  271 

wave  in  one,  the  other  in  the  other  transformer  coil  —  is  half  this 
value,  or 


=  7?  V  —  ~<r  I  /      an*^  +  f  "[sin  #"  +  si 

2  71      —     U      (   ^  0  JQ 


"  +  sin  (^  -  6i 


4V^sf-^'+Asin0'sm(0 

2    ?    71     -    01    (  J0  J0 

V       /        1          7~I        r~~  Y1 

=  -2y-    -r-)^  +  tf'cos^  -sin  (2^  -  6m 
2      TT  —  0t  ( 2      L  Jo 


-  +  .0t  cos  0X  -  sin  0j  > ; 


or,  substituting 


^2    =    *0 


2 


-0,)        +  0,  cos  0,  -  sin  0, 


or,  substituting 


i<  —  ~~ ~        IT  v  -L  H~  cos  sin 


2  2V2 

where —  =  ratio  of  effective  value  to  mean  value  of  sine  wave. 

2V2 

27.  An  approximate  representation  by  equivalent  sine  waves, 
if  e0  =  the  mean  value  of  direct  terminal  voltage,  i0.=  the  mean 
value  of  direct  current,  is  therefore  as  follows: 

The  secondary  generated  e.m.f.  of  the  transformer  is 


272  TRANSIENT  PHENOMENA 

the  secondary  current  of  the  transformer  is 


2  < 
the  pulsation  of  the  direct  current  is 

7T2 


cos 


sin 


x  +  60       Ti  +  6, 


the  anode  voltage  of  the  rectifier  is 


2  #0  -  sin  2  #c 


and  herefrom  follows  the  apparent  efficiency  of  rectification,  -V 

61 

the  power  factor,  the  efficiency,  etc. 


70°       80° 


Fig.  79.    E.m.f.  and  current  ratio  and  secondary  power  factor  of  constant- 
current  mercury  arc  rectifier. 

From  the  equivalent  sine  waves,  e  and  i,  of  the  transformer 
secondary,  and  their  phase  angle,  the  primary  impressed  e.m.f. 
and  the  primary  current  of  the  transformer,  and  thereby  the 


ARC  RECTIFICATION  273 

power  factor,  the  efficiency,  and  the  apparent  efficiency  of  the 
system,  are  calculated  in  the  usual  manner. 

In  the  secondary  circuit,  the  power  factor  is  below  unity 
essentially  due  to  wave  shape  distortion,  less  due  to  lag  of  cur- 
rent. 

As  example  are  shown,  in  Fig.  79,  with  the  angle  of  overlap  00 

e  2i 

as  abscissas,  the  ratio  of  voltages,  - — ;  the  ratio  of  currents,  — ; 

2eo  % 

i' 
the  current  pulsation,  ->  and  the  power  factor  of  the  secondary 

to 
circuit. 


SECTION  III 
TRANSIENTS   IN   SPACE 


TRANSIENTS   IN   SPACE 


CHAPTER  I. 

INTRODUCTION. 

1.  The  preceding  sections  deal  with  transient  phenomena  in 
time,  that  is,  phenomena  occurring  during  the  time  when  a 
change  or  transition  takes  place  between  one  condition  of  a  cir- 
cuit and  .another.  The  time,  t,  then  is  the  independent  variable, 
electric  quantities  as  current,  e.m.f.,  etc.,  the  dependent  variables. 

Similar  transient  phenomena  also  occur  in  space,  that  is,  with 
space,  distance,  length,  etc.,  as  independent  variable.  Such 
transient  phenomena  then  connect  the  conditions  of  the  electric 
quantities  at  one  point  in  space  with  the  electric  quantities  at 
another  point  in  space,  as,  for  instance,  current  and  potential 
difference  at  the  generator  end  of  a  transmission  line  with  those 
at  the  receiving  end  of  the  line,  or  current  density  at  the  surface 
of  a  solid  conductor  carrying  alternating  current,  as  the  rail 
return  of  a  single-phase  railway,  with  the  current  density  at  the 
center  or  in  general  inside  of  the  conductor,  or  the  distribution 
of  alternating  magnetism  inside  of  a  solid  iron,  as  a  lamina  of  an 
alternating-current  transformer,  etc.  In  such  transient  phenom- 
ena in  space,  the  electric  quantities,  which  appear  as  functions 
of  space  or  distance,  are  not  the  instantaneous  values,  as  in  the 
preceding  chapters,  but  are  alternating  currents,  e.m.fs.,  etc., 
characterized  by  intensity  and  phase,  that  is,  they  are  periodic 
functions  of  time,  and  the  analytical  method  of  dealing  with 
such  phenomena  therefore  introduces  two  independent  variables, 
time  t  and  distance  I,  that  is,  the  electric  quantities  are  periodic 
functions  of  time  and  transient  functions  of  space. 

The  introduction  of  the  complex  quantities,  as  representing  the 
alternating  wave  by  a  constant  algebraic  number,  eliminates 

277 


278  TRANSIENT    PHENOMENA 

the  time  t  as  variable,  so  that,  in  the  denotation  by  complex 
quantities,  the  transient  phenomena  in  space  are  functions  of 
one  independent  variable  only,  distance  Z,  and  thus  lead  to  the 
same  equations  as  the  previously  discussed  phenomena,  with 
the  difference,  however,  that  here,  in  dealing  with  space  phenom- 
ena, the  dependent  variables,  current,  e.m.f.,  etc.,  are  complex 
quantities,  while  in  the  previous  discussion  they  appeared  as 
instantaneous  values,  that  is,  real  quantities. 

Otherwise  the  method  of  treatment  and  the  general  form  of 
the  equations  are  the  same  as  with  transient  functions  of  time. 

2.  Some  of  the  cases  in  which  transient  phenomena  in  space 
are  of  importance  in  electrical  engineering  are : 

(a)  Circuits  containing  distributed  capacity  and  self-induc- 
tance, as  long-distance  energy  transmission  lines,  long-distance 
telephone  circuits,  multiple  spark-gaps,  as  used  in  some  forms 
of  high  potential  lightning  arresters  (multi-gap  arrester),  etc. 

(b)  The  distribution  of  alternating  current  in  solid  conductors 
and  the  increase  of  effective  resistance  and  decrease  of  effective 
inductance  resulting  therefrom. 

(c)  The  distribution  of  alternating  magnetic  flux  in  solid  iron, 
or  the  screening  effect  of  eddy  currents  produced  in  the  iron,  and 
the  apparent  decrease  of  permeability  and  increase  of  power 
consumption  resulting  therefrom. 

(d)  The  distribution  of   the  electric   field    of    a   conductor 
through  space,  resulting  from  the  finite  velocity  of  propagation 
of  the  electric  field,  and  the  variation  of  self-inductance  and 
mutual  inductance  and  of  capacity  of  a  conductor  without 
return,  as  function  of  the  frequency,  in  its  effect  on  wireless 
telegraphy. 

(e)  Conductors  conveying  very  high  frequency  currents,  as 
lightning  discharges. 


CHAPTER  II. 

LONG-DISTANCE   TRANSMISSION   LINE. 

3.  If  an  electric  impulse  is  sent  into  a  conductor,  as  a  trans- 
mission line,  this  impulse  travels  along  the  line  at  the  velocity 
of  light  (approximately),  or  188,000  miles  per  second.  If  the 
line  is  open  at  the  other  end,  the  impulse  there  is  reflected  and 
returns  at  the  same  velocity.  If  now  at  the  moment  when  the 
impulse  arrives  at  the  starting  point  a  second  impulse,  of 
opposite  direction,  is  sent  into  the  line,  the  return  of  the  first 
impulse  adds  itself,  and  so  increases  the  second  impulse;  the 
return  of  this  increased  second  impulse  adds  itself  to  the  third 
impulse,  and  so  on;  that  is,  if  alternating  impulses  succeed  each 
other  at  intervals  equal  to  the  time  required  by  an  impulse  to 
travel  over  the  line  and  back,  the  effects  of  successive  impulses 
add  themselves,  and  large  currents  and  high  e.m.fs.  may  be 
produced  by  small  impulses,  that  is,  low  impressed  alternating 
e.m.fs.,  or  inversely,  when  once  started,  even  with  zero  impressed 
e.m.f.,  such  alternating  currents  traverse  the  lines  for  some  time, 
gradually  decreasing  in  intensity  by  the  energy  consumption  in 
the  conductor,  and  so  fading  out. 

The  condition  of  this  phenomenon  of  electrical  resonance 
thus  is  that  alternating  impulses  occur  at  time  intervals  equal 
to  the  time  required  for  the  impulse  to  travel  the  length  of  the 
line  and  back;  that  is,  the  time  of  one  half  wave  of  impressed 
e.m.f.  is  the  time  required  by  light  to  travel  twice  the  length  of 
the  line,  or  the  time  of  one  complete  period  is  the  time  light 
requires  to  travel  four  times  the  length  of  the  line;  in  other 
words,  the  number  of  periods,  or  frequency  of  the  impressed 
alternating  e.m.fs.,  in  resonance  condition,  is  the  velocity  of 
light  divided  by  four  times  the  length  of  the  line;  or,  in  free 
oscillation  or  resonance  condition,  the  length  of  the  line  is  one 
quarter  wave  length. 

279 


280  TRANSIENT  PHENOMENA 

If  then  I  =  length  of  line,  S  =  speed  of  light,  the  frequency  of 
oscillations  or  natural  period  of  the  line  is 

— 
'•  "4? 

or,  with  I  given  in  miles,  hence  S  =  188,000  miles  per  second,  it  is 

,       47,000 
/o  =  — j-    cycles.  (2) 

To  get  a  resonance  frequency  as  low  as  commercial  frequencies, 
as  25  or  60  cycles,  would  require  Z  ==  1880  miles  for  /0  =  25 
cycles,  and  Z  =  783  miles  for./,  -  60  cycles. 

It  follows  herefrom  that  many  existing  transmission  lines  are 
such  small  fractions  of  a  quarter-wave  length  of  the  impressed 
frequency  that  the  change  of  voltage  and  current  along  the  line 
can  be  assumed  as  linear,  or  at  least  as  parabolic;  that  is,  the  line 
capacity  can  be  represented  by  a  condenser  in  the  middle  of  the 
line,  or  by  condensers  in  the  middle  and  at  the  two  ends  of  the 
line,  the  former  of  four  times  the  capacity  of  either  of  the  two 
latter  (the  first  approximation  giving  linear,  the  second  a  para- 
bolic distribution). 

For  further  investigation  of  these  approximations  see  "Theory 
and  Calculation  of  Alternating-Current  Phenomena/'  4th  edition, 
pages  225  to  233. 

If,  however,  the  wave  of  impressed  e.m.f.  contains  appreciable 
higher  harmonics,  some  of  the  latter,  may  approach  resonance 
frequency  and  thus  cause  trouble.  For  instance,  with  a  line  of 
150  miles  length,  the  resonance  frequency  is  /0  =  313  cycles  per 
second,  or  between  the  5th  harmonic  and  the  7th  harmonic,  300 
and  420  cycles  of  a  60-cycle  system;  fairly  close  to  the  5th  har- 
monic. 

The  study  of  such  a  circuit  of  distributed  capacity  thus 
becomes  of  importance  with  reference  to  the  investigation  of 
the  effects  of  higher  harmonics  of  the  generator  wave. 

In    long-distance    telephony   the    important    frequencies    of 

speech  probably  range  from  100  to  2000  cycles.     For  these  fre- 
er 

quencies  the  wave  length  varies  from  —  =  1880  miles  down  to 

L 

94  miles,  and  a  telephone  line  of  1000  miles  length  would  thus 


LONG-DISTANCE  TRANSMISSION  LINE  281 

contain  from  about  one-half  to  11  complete  waves  of  the  im- 
pressed frequency.  For  long-distance  telephony  the  phenomena 
occurring  in  the  line  thus  can  be  investigated  only  by  consider- 
ing the  complete  equation  of  distributed  capacity  and  inductance 
as  so-called  "wave  transmission"  and  the  phenomena  thus 
essentially  differ  from  those  in  a  short  energy  transmission  line. 

4.  Therefore  in  very  long  circuits,  as  in  lines  conveying  alter- 
nating currents  of  high  value  at  high  potential  over  extremely 
long  distances,  by  overhead  conductors  or  underground  cables, 
or  with  very  feeble  currents  at  extremely  high  frequency,  such 
as  telephone  currents,  the  consideration  of  the  line  resistance, 
which  consumes  e.m.fs.  in  phase  with  the  current,  and  of  the 
line  reactance,  which  consumes  e.m.fs.  in  quadrature  with  the 
current,  is  not  sufficient  for  the  explanation  of  the  phenomena 
taking  place  in  the  line,  but  several  other  factors  have  to  be  taken 
into  account. 

In  long  lines,  especially  at  high  potentials,  the  electrostatic 
capacity  of  the  line  is  sufficient  to  consume  noticeable  currents. 
The  charging  current  of  the  line  condenser  is  proportional  to  the 
difference  of  potential  and  is  one-fourth  period  ahead  of  the 
e.m.f.  Hence,  it  either  increases  or  decreases  the  main  current, 
according  to  the  relative  phase  of  the  main  current  and  the  e.m.f. 

As  a  consequence  the  current  changes  in  intensity,  as  well  as 
in  phase,  in  the  line  from  point  to  point;  and  the  e.m.fs.  con- 
sumed by  the  resistance  and  inductance,  therefore,  also  change 
in  phase  and  intensity  from  point  to  point,  being  dependent 
upon  the  current. 

Since  no  insulator  has  an  infinite  resistance,  and  since  at  high 
potentials  not  only  leakage  but  even  direct  escape  of  electricity 
into  the  air  takes  place  by  "  brush  discharge,"  we  have  to  rec- 
ognize the  existence  of  a  current  approximately  proportional 
and  in  phase  with  the  e.m.f.  of  the  line.  This  current  represents 
consumption  of  power,  and  is  therefore  analogous  to  the  e.m.f. 
consumed  by  resistance,  while  the  condenser  current  and  the 
e.m.f.  of  inductance  are  wattless  or  reactive. 

Furthermore,  the  alternating  current  passing  over  the  line  pro- 
duces in  all  neighboring  conductors  secondary  currents,  which 
react  upon  the  primary  current  and  thereby  introduce  e.m.fs. 
of  mutual  inductance  into  the  primary  circuit.  Mutual  induc- 
tance is  neither  in  phase  nor  in  quadrature  with  the  current, 


282  TRANSIENT  PHENOMENA 

and  can  therefore  be  resolved  into  a  power  component  of  mutual 
inductance  in  phase  with  the  current,  which  acts  as  an  increase 
of  resistance,  and  into  a  reactive  component  in  quadrature  with 
the  current,  which  decreases  the  self-inductance. 

This  mutual  inductance  is  not  always  negligible,  as,  for 
instance,  its  disturbing  influence  in  telephone  circuits  shows. 

The  alternating  potential  of  the  line  induces,  by  electrostatic 
influence,  electric  charges  in  neighboring  conductors  outside  of 
the  circuit,  which  retain  corresponding  opposite  charges  on  the 
line  wires.  This  electrostatic  influence  requires  the  expenditure 
of  a  current  proportional  to  the  e.m.f.  and  consisting  of  a 
power  component  in  phase  with  the  e.m.f.  and  a  reactive  com- 
ponent in  quadrature  thereto. 

The  alternating  electromagnetic  field  of  force  set  up  by  the 
line  current  produces  in  some  materials  a  loss  of  power  by  mag- 
netic hysteresis,  or  an  expenditure  of  e.m.f.  in  phase  with  the  cur- 
rent, which  acts  as  an  increase  of  resistance.  This  electro- 
magnetic hysteresis  loss  may  take  place  in  the  conductor  proper 
if  iron  wires  are  used,  and  may  then  be  very  serious  at  high  fre- 
quencies such  as  those  of  telephone  currents. 

The  effect  of  eddy  currents  has  already  been  referred  to  under 
"  mutual  inductance,"  of  which  if  is  a  power  component. 

The  alternating  electrostatic  field  of  force  expends  power  in 
dielectrics  by  what  is  called  dielectric  hysteresis.  In  concentric 
cables,  where  the  electrostatic  gradient  in  the  dielectric  is  com- 
paratively large,  the  dielectric  hysteresis  may  at  high  potentials 
consume  considerable  amounts  of  power.  The  dielectric  hystere- 
sis appears  in  the  circuit  as  consumption  of  a  current  whose 
component  in  phase  with  the  e.m.f.  is  the  dielectric  power  current, 
which  may  be  considered  as  the  power  component  of  the  charging 
current. 

Besides  this  there  is  the  apparent  increase  of  ohmic  resistance 
due  to  unequal  distribution  of  current,  which,  however,  is  usually 
not  large  enough  to  be  noticeable  at  low  frequencies. 

Also,  especially  at  very  high  frequency,  energy  is  radiated  into 
space,  due  to  the  finite  velocity  of  the  electric  field,  and  can  be 
represented  by  power  components  of  current  and  of  voltage 
respectively. 

5.  This  gives,  as  the  most  general  case  and  per  unit  length 
of  line, 


LONG-DISTANCE  TRANSMISSION  LINE  283 

E.m.fs.  consumed  in  phase  with  the  current,  I,  and  =  r/,  repre- 
senting consumption  of  power,  and  due  to  resistance,  and  its 
apparent  increase  by  unequal  current  distribution;  to  the  power 
component  of  mutual  inductance:  to  secondary  currents;  to  the 
power  component  of  self  -inductance:  to  electromagnetic  hysteresis; 
and  to  electromagnetic  radiation. 

E.m.fs.  consumed  in  quadrature  with  the  current,  I,  and  =  xl, 
reactive,  and  due  to  self-inductance  and  mutual  inductance. 

Currents  consumed  in  phase  with  the  e.m.f.,  E,  and  =  gE, 
representing  consumption  of  power,  and  due  to  leakage  through 
the  insulating  material,  including  brush  discharge;  to  the  power 
component  of  electrostatic  influence;  to  the  power  component  of 
capacity,  or  dielectric  hysteresis,  and  to  electrostatic  radiation. 

Currents  consumed  in  quadrature  with  tJw  e.m.f.,  E,  and  =  bE, 
being  reactive,  and  due  to  capacity  and  electrostatic  influence. 

Hence  we  get  four  constants  per  unit  length  of  line,  namely: 
Effective  resistance,  r;  effective  reactance,  x;  effective  conduc- 
tance, g,  and  effective  susceptance,  b  =  -  bc  (bc  being  the 
absolute  value  of  susceptance).  These  constants  represent  the 
coefficients  per  unit  length  of  line  of  the  following:  e.m.f. 
consumed  in  phase  with  the  current;  e.m.f.  consumed  in  quadra- 
ture with  the  current;  current  consumed  in  phase  with  the  e.m.f., 
and  current  consumed  in  quadrature  with  the  e.m.f. 

6.  This  line  we  may  assume  now  as  supplying  energy  to  a 
receiver  circuit  of  any  description,  and  determine  the  current  and 
e.m.f.  at  any  point  of  the  circuit. 

That  is,  an  e.m.f.  and  current  (differing  in  phase  by  any 
desired  angle)  may  be  given  at  the  terminals  of  the  receiving 
circuit.  To  be  determined  are  the  e.m.f.  and  current  at  any 
point  of  the  line,  for  instance,  at  the  generator  terminals;  or 
the  impedance,  Zt  =  rl  -  jxv  or  admittance,  Yl  =  g1  +  jblt 
of  the  receiver  circuit,  and  e.m.f.,  E0,  at  generator  terminals  are 
given;  the  current  and  e.m.f.  at  any  point  of  circuit  to  be  deter- 
mined, etc. 

7.  Counting-  now  the  distance,  I,  from  a  point  0  of  the  line 
which  has  the  e.m.f. 


+  je,f 


and  the  current 


284  TRANSIENT  PHENOMENA 

and  counting  I  positive  in  the  direction  of  rising  power  and 
negative  in  the  direction  of  decreasing  power,  at  any  point  I,  in 
the  line  differential  dl  the  leakage  current  is 

Egdl 
and  the  capacity  current  is 

-  jEb  dl; 

hence,  the  total  current  consumed  by  the  line  differential  dl  is 
dl  =  E  (g  -  jb)  dl 
=  EY  dl, 

%-YE.  (1) 

In  the  line  differential  dl  the  e.m.f.  consumed  by  resistance  is 

Irdl, 

the  e.m.f.  consumed  by  inductance  is 

-  jlxdl; 

hence,  the  total  e.m.f.  consumed  by  the  line  differential  dl  is 

dE  =  I  (r  -  jx)  dl 
=  IZ  dl, 

f-Zl.  (2) 

These  fundamental  differential  equations  (1)  and  (2)  are  sym- 
metrical with  respect  to  /  and  E. 

Differentiating  these  equations  (1)  and  (2)  gives 


dl  (3) 

d*E 
and 


LONG-DISTANCE   TRANSMISSION  LINE  285 

and  substituting  (1)  and  (2)  in  (3)  gives  the  differential  equa- 
tions of  E  and  I  ,  thus  : 

%-YZB  (4) 

and  ;r=FZ--  (5) 

These  differential  equations  are  identical,  and  consequently  I  and 
E  are  functions  differing  by  their  integration  constants  or  by  their 
limiting  conditions  only. 

These  equations  are  of  the  form 

f  - 

and  are  integrated  by 

V  - 

where  e  is  the  basis  of  the  natural  logarithms,  =  2.718283. 
Choosing  equation  (5),  which  is  integrated  by 

7  --  At™,  (6) 

and  differentiating  (6)  twice  gives 


and  substituting  (6)  in  (5),  the  factor  Asvl  cancels,  and  we  have 

V2  =  ZY, 
or 

V  -=  VZY,  (7) 

hence,  the  general  integral, 

/  =  A,e  +  vl  -  A2£-vl.  (8) 

By  equation  (1), 

E      ld' 
Y~dl' 

and  substituting  herein  equation  (8)  gives 

E  -A,™  +  A*-"     ,  (9) 


286  TRANSIENT  PHENOMENA 

or,  substituting  (7), 

E  =\/A1e+vl+A,e-vi      .  (10) 


The  integration  constants  A1  and  A2  in  (8),  (9),  (10),  in 
general,  are  complex  quantities.  The  coefficient  of  the  exponent, 
F,  as  square  root  of  the  product  of  two  complex  quantities,  also 
is  a  complex  quantity,  therefore  may  be  written 

V  =  a  -  jp,  (11) 

and  substituting  for  F,  Z  and  Y  gives 

(a  -  j/?)2  =  (r  -  jx)  (g  -  jb), 
or 

(a2  -  /?2)  -  2  jap  =  (rg  -  xb)  -  j  (rb  +  gx), 

and  this  resolves  into  the  two  separate  equations 

a2  —  ft2  =  rg  —  xb ) 

2  aQ  =  rb  +        \  (12) 

since,  when  two  complex  quantities  are  equal,  their  real  terms  as 
well  as  their  imaginary  terms  must  be  equal. 
Equations  (12)  sauared  and  added  give 

(a2  +  /?2)2  =  (rg  -  xb)2  +  (rb  +  xg)* 
=  (r2  +  z2)  tf  +  62) 


hence, 

«2  +  P  =  *0,  (13) 

and  from  (12)  and  (13), 


a  =  \^(zy  +  rg  -  xb) 

and  (14) 

ft  =  V%(zy  -  rg  +  xb).  I 


(15) 


Equations  (8)  and  (10)  now  assume  the  form 
and 


E  = 


LONG-DISTANCE  TRANSMISSION  LINE  287 

Substituting  for  the  exponential  function  with  an  imaginary 
exponent  the  trigonometric  expression 

e±m  -  cos  02  ±  j  sin  pi,  (16) 

equations  (15)  assume  the  form 
/  =  4l£+a'(cos0/  -  /sin  /#)  -  42£~a'(cos0Z  +  /  sin  pi) 


/z( 

E  =\  -\Af  +al(cos  pi  -  /  sin  00  +42e  ~  a'(cos  pi  +/  sin  00 


where  Al  and  A2  are  the  constants  of  integration. 

The  distribution  of  current  /  and  voltage  E  along  the  circuit, 
therefore,  is  represented  by  the  sum  of  two  products  of  expo- 
nential and  trigonometric  functions  of  the  distance  I.  Of  these 
terms,  the  one,  with  factor  As+al,  increases  with  increasing  dis- 
tance /,  that  is,  increases  towards  the  generator,  while  the  other, 
with  factor  A^-*1 ,  decreases  towards  the  generator  and  thus 
increases  with  increasing  distance  from  the  generator.  The 
phase  angle  of  the  former  decreases,  that  of  the  latter  increases 
towards  the  generator,  and  the  first  term  thus  can  be  called  the 
main  wave,  the  second  term  the  reflected  wave. 

At  the  point  /  =  0,  by  equations  (17)  we  have 

7        -   A  A 

*  o    "~    .1  ~~  *lv 


and  the  ratio 

^ 

-  =  m  (cos  T  +  /  sin  r), 
Ai 

where  r  may  be  called  the  angle  of  reflection,  and  m  the  ratio  of 
amplitudes  of  reflected  and  main  wave  at  the  reflection  point. 

8.  The  general  integral  equations  of  current  and  voltage  dis- 
tribution (17)  can  be  written  in  numerous  different  forms. 

Substituting  —  A2  instead  of  +  A2,  the  sign  between  the 
terms  reverses,  and  the  current  appears  as  the  sum,  the  voltage 
as  difference  of  main  and  reflected  wave. 


288 


TRANSIENT  PHENOMENA 


Rearranging  (17)  gives 

/  =  (Aie+t*  -  A2e~al)  cos  pi  -  j  (Ale+al+  A2e~al)  sin/?/ 
and 


E  == 


Substituting  (7)  gives 


and  substituting 


and 


or 


and 


(18) 


=  --- 
Y      V      Y' 


(19) 


*-  B 

"  -v 


changes  equations  (17)  to  the  forms, 

7=7  \Ble+al(co8pl-j  sin  pi)  -B^'"1  (cos  pi +j  sin.pl)  I 


and 


or 


f  +al(cos  pl-j  sin 


al  (cos  ^  +j  sin  pi)  , 


(20) 


I  =  Y  \  C^  +aZ  (cos  pi — j  sin  pi)  -  C2e   al  (cos  pi  +j  sin  pi) 
>  +aZ(cos  pl-j  sin  pi)  +C2e~ 


I  = 
and 


sn 


(21) 


LONG-DISTANCE  TRANSMISSION  LINE  289 


Substituting  in  (17) 


4, 


VY  VY 

gives 


-D2£-aZ(cos/?Z+/sin  0Z) 


and 


(23) 


(22) 


Reversing  the  sign  of  Z,  that  is,  counting  the  distance  in  the 
opposite  direction,  or  positive  for  decreasing  power,  from  the 
generator  towards  the  receiving  circuit,  and  not,  as  in  equations 
(17)  to  (22),  from  the  receiving  circuit  towards  the  generator, 
exchanges  the  position  of  the  two  terms;  that  is,  the  first  term, 
or  the  main  wave,  decreases  with  increasing  distance,  and  lags; 
the  second  term,  or  the  reflected  wave,  increases  with  the  dis- 
tance, and  leads. 

Equations  (47)  thus  assume  the  form 
/  -=  4l£-a<  (cos  pi  +  j  sin  pi)  -  A2e+ai  (cos  pi  -  j  sin  pi) 
and 

fz  ( 

E=  y  -  J41e-rf(cos0Z  +  /sin0Z)  +  A2s+al(cospl-jsinpl) 

and  correspondingly  equations  (18)  to  (22)  modify. 

9.  The  two  integration  constants  contained  in  equations  (17) 
to  (23)  require  two  conditions  for  their  determination,  such  as 
current  and  voltage  at  one  point  of  the  circuit,  as  at  the  generator 
or  at  the  receiving  end;  or  current  at  one  point,  voltage  at  the 
other;  or  voltage  at  one  point,  as  at  the  generator,  and  ratio  of 
voltage  and  current  at  the  other  end,  as  the  impedance  of  the 
receiving  circuit. 

Let  the  current  and  voltage  (in  intensity  as  well  as  phase,  that 
is,  as  complex  quantities)  be  given  at  one  point  of  the  circuit, 
and  counting  the  distance  Z  from  this  point,  the  terminal  con- 
ditions are 

2-0, 

r-  '{•  -  s'  +  ?v> 

and  E  =  E0  =  e0  +  jeQ'. 


(24) 


290  TRANSIENT  PHENOMENA 

Substituting  (24)  in  (17)  gives 


and 


hence, 


and 


=  V/f  (4  +  4,); 


and  substituted  in  (17)  gives 

_  1 
•  ~2 


-  jsin/?/) 
+  /sin/?/) 


and 


0  -  /0  V       E- 


(cos  /?Z 


sn 


(25) 


If  then  70  and  ^Oare  the  current  and  voltage  respectively  at 
the  receiving  end  or  load  end  of  a  circuit  of  length  1Q,  equations 
(25)  represent  current  and  voltage  at  any  point  of  the  circuit, 
from  the  receiving  end  Z  =  0  to  the  generator  end  I  =  10. 

If  70  and  E0  are  the  current  and  voltage  at  the  generator 
terminals,  since  in  equations  (17)  I  is  counted  towards  rising 
power,  in  the  present  case  the  receiving  end  of  the  line  is  repre- 
sented by  I  =  —  Z0;  that  is,  the  negative  values  of  I  represent 
the  distance  from  the  generator  end,  along  the  line.  In  this 
case  it  is  more  convenient  to  reverse  the  sign  of  /,  that  is,  use 
equations  (22)  and  the  distribution  of  current  and  voltage  at 
distance  I  from  the  generator  terminals.  7 ,  E  are  then  given  by 


LONG-DISTANCE  TRANSMISSION  LINE 


291 


•  =  \ 


and 


v  f  )*~ 

o  V  |)  *  +^ 

o  V  f)£~a' 


(cos  fti  +  j  sin 


~     Sn 


(26) 


10.  Assume  that  the  character  of  the  load,  that  is,  the  impe- 

771  T 

dance,  — -1  =Z.=r,—/jx.,  or  admittance, 77  =F=  —  =  &+$*, 

Ti  •  i  i 

of  the  receiving  circuit  and  the  voltage  E0  at  the  generator  end 

of  the  circuit  be  given. 

Let  /0  =  length  of  circuit,  and  counting  distance  I  from  the 
generator  end,  for  I  =  0  we  have 

E  =  E- 


this  substituted  in  equation  (23)  gives 


However,  for  I 


(27) 


E      7  . 
J  =  Z» 


substituting  (23)  herein  gives 
Z  =  y/^  4  te  -  ^  (cos  fflp  +  j  sin 


(cos  ^0  -  j  sin  ffl0)  § 


F  AlS-*  (cos/?/0+  j  sin/?Z0)-  4/+afo  (cos/?Z0-  /si 
hence,  substituting  (19)  and  expanding, 


or 


A    VZ*~Z:- 

—  ^A«  • —c 


VZ   -  Z' 


42  -  4,  vzl          ';"°  (cos  2  /?/.  +  /  sin 


292  TRANSIENT  PHENOMENA 

and  denoting  the  complex  factor  by 

V7     -  7 

C  =        l          t  -2  "'"(cos  2  pi.  +  j  sin  fa),  (28) 


4,  =  QAV 


which  may  be  called  the  reflection  constant,  we  have 
and  by  (27), 

4j  =  — 

K<>     ~IY  (29) 

and 

hence,  substituted  in  (22), 

/  = 

and 

E  =      -°     U~a/  (cos  /?Z  +  y  sin  /?/)  +  Cs  +aZ  (cos  /M  -  j  sin  /?Z) 

(30) 

11.  As  an  example,  consider  the  problem  of  delivering,  in  a 
three-phase  system,  200  amperes  per  phase,  at  90  per  cent  power 
factor  lag  at  60,000  volts  per  phase  (or  between  line  and  neutral) 
and  60  cycles,  at  the  end  of  a  transmission  line  200  miles  in 
length,  consisting  of  two  separate  circuits  in  multiple,  each 
consisting  of  number  00  B.  and  S.  wire  with  6  feet  distance 
between  the  conductors. 

Number  00  B.  and  S.  wire  has  a  resistance  of  0.42  ohms  per 
mile,  and  at  6  feet  distance  from  the  return  conductor  an 
inductance  of  2.4  mh.  and  capacity  of  0.015  mf.  per  mile. 

The  two  circuits  in  multiple  give,  at  60  cycles,  the  following 
line  constants  per  mile:  r  =  0.21  ohm,  L  =  1.2  X  10" 3  henry, 
and  C  =  0.03  X  10" 6  farad;  hence, 

x  =  2  TT/L  =  0.45, 

z  =  0.21  -  0.45  y, 

z   =  0.50, 
and,  neglecting  the  conductance  (g  =  0), 

b  =  27T/C  =  IIXHT8, 

Y  =  -- 11  x  10- 6y, 

y   =  11  X  10~6, 


and 


and 


LONG-DISTANCE  TRANSMISSION  LINE 


a  =  0.524  X  10~ 3, 
P  **  2.285  X  10" 3, 
V  =  (0.524  -  2.285?)  10- 3, 

=  ^  =  (4.53  -  0.9  j)  10~3 
Zi 

p  =  ^=  (0.208  +  0.047  /)  10 +  3. 


293 

(31) 


Counting  the  distance  I  from  the  receiving  end,  and  choosing 
the  receiving  voltage  as  zero  vector,  we  have 


E  =  E0  =  e0  =  60,000  volts, 

and  the  current  of  200  amperes  at  90  per  cent  power  factor, 

87  j, 


and  substituting  these  values  in  equations  (25)  gives 

/  =  (226  +  14.4  j)  e+al  (cos  pl-j  sin  pi)  -  (46-72.6  j)  e~al 


(cos  ftl  +  j  sin  /#),  in  amperes, 


and 

£"=  (46.7  + 13.3  ?>+aZ  (cos  pi-  /sin 


(13.3-  13.3  j) 


(32) 


(cos  /?Z  +  j  sin  /?/),  in  kilo  volts, 

where  a  and  /?  are  given  by  above  equations  (31). 

From  equations  (32)  the  following  results  are  obtained. 


Receiving  end  of  line, 
7  =  180  +  87  / 
E=    60  X  103 

Middle  of  line, 

7  =  177+  18  / 


I  =0 

i  =  200  amp. 

e  =  60,000  volts 

I  =  100 

i  =  178  amp. 


E=  (66.2  -  6.9 /)  103        e  =  66,400  volts 


Generator  end  of  line, 
/=  165.7  -  56  / 
E=  (69  -  15 /)  103 


I  =  200 

i  =  175  amp. 

e  =  70,700  volts 


tan  0,  =  0.483  6,  =  26° 

0,=    0 
power  factor,    0.90  lag 

tan  6l=  +  0.102     6l  =      6° 
tan  02  =-0.104     62  =  -  6° 
d,  -  02  =J~^     12° 
power  factor,  cos  6  =  0.979  lag. 

tan0t  =-0.338     0,  =  -19° 
tan  02  =  - 0.218     02  =  -12° 
Ol  -  02  =  0  =  -  7° 
power  factor,  cos  0  =  0.993  lead. 


294  TRANSIENT  PHENOMENA 

As  seen,  the  current  decreases  from  the  receiving  end  to  the 
middle  of  the  line,  but  from  there  to  the  generator  remains  prac- 
tically constant.  The  voltage  increases  more  in  the  receiving 
half  of  the  line  than  in  the  generator  half.  The  power-factor  is 
practically  unity  from  the  middle  of  the  line  to  the  generator. 

12.  It  is  interesting  to  compare  with  above  values  the  values 
derived  by  neglecting  the  distributed  character  of  resistance, 
inductance,  and  capacity. 

From  above  constants  per  mile  it  follows,  for  the  total  line  of 
200  miles  length,  r0  =  42  ohms,  x0  =  90  ohms,  and  60  =  2.2 
X  10~3  mho;  hence, 

Z0  =  42  -  90  / 
and  F0  =  -  2.2  j  10-  3. 

(1)  Neglecting  the  line  capacity  altogether,  with  70  and  EQ  at 
the  receiver  terminals,  at  the  generator  terminals  we  have 


and 


hence, 

/!  =  180  +  87;  t\  =  200  amp.  tan  01  =      0.483    6l  =+  26° 

E,=  (75.4  -  12.6/)  103        e,  =  76,400  volts        tan  62  =  -  0.167    62  =    -    9° 

Ol  -  62  =  d   =+  34° 
power  factor,  cos  6   =  0.83  lag. 

These  values  are  extremely  inaccurate,  voltage  and  current 
at  generator  too  high  and  power  factor  too  low. 

(2)  Representing  the  line  capacity  by  a  condenser  at  the 
generator  end,  that  is,  adding  the  condenser  current  at  the 
generator  end, 

(»  -  f.->  y0?i 

and 

^i  =  ^o  +  ^o(0; 
hence, 

7,  =  152  -  89  /  t,  =  176  amp.  tan  0l  =-  0.585    0t  =  -30° 

E,=  (75  A  -  12.6  /)103     el  =  76,400  volts  tan  02  =  -0.167    02  =    -  9° 

0,  -  02  =  e  =-  2iQ 

power  factor,  cos  0    =  0.93  lead. 


LONG-DISTANCE  TRANSMISSION  LINE  295 

As  seen,  the  current  is  approximately  correct,  but  the  voltage 
is  far  too  high  and  the  power  factor  is  still  low,  but  now  leading. 

(3)  Representing  the  line   capacity  by  a  condenser  at  the 
receiving  end,  that  is,  adding  the  condenser  current  at  the  load, 

/,  =  7.  +  FA 

and 

El  =  Et  +  ZtIl; 

hence, 

7,  =  180  -  45  /  t\  =  186  amp.  tan  6,=-  .250    0l  =  - 14° 

EI  =  (63.5  -  18.1  /)  103         el  =  66,000  volts          tan  02  =  -  .285    62  =  -16° 

6l-62  =  6  =  +  2° 
power  factor,  cos  6  =    1.00 

In  this  case  the  voltage  el  is  altogether  too  low,  the  current 
somewhat  high,  but  the  power  factor  fairly  correct. 

(4)  Taking  the  average  of  the  values  of  (2)  and  of  (3)  gives 

/,  =  166  -  67 /  i\  =  179  amp.  tan  0l  =  -  0.403    0l  =  -  22° 

El  =  (69.4  -  15.3  /)  103         el  =  71,100  volts       tan  0a  =  -  0.220    02  =  -  12° 

01  -  62  =  0   =  -  10° 
power-factor,  cos  6   =  0.985  lead. 

As  seen  by  comparing  these  average  values  with  the  exact 
result  as  derived  above,  these  values  are  not  very  different,  but 
constitute  a  fair  approximation  in  the  present  case.  Such  a 
close  coincidence  of  this  approximation  with  the  exact  result  can, 
however,  not  be  counted  upon  in  all  instances. 

13.   In  the  equations  (17)  to  (23)  the  length 

1.-J-  (33) 

2x 
is  a  complete  wave  length,  which  means  that  in  the  distance  — 

the  phases  of  the  components  of  current  and  of  e.m.f.  repeat,  and 
that  in  half  this  distance  they  are  just  opposite. 

Hence,  the  remarkable  condition  exists  that  in  a  very  long 
line  at  different  points  the  currents  are  simultaneously  in  oppo- 
site directions  and  the  e.m.fs.  are  opposite. 


296  TRANSIENT  PHENOMENA 

The  difference  of  space  phase  T  between  current  I  and  e.m.f. 
E  at  any  point  I  of  the  line  is  determined  by  the  equation 

Tjl 

m  (cos  r  +  j  sin  r)  =  j  >  (34) 

where  w  is  a  constant. 

Hence,  r  varies  from  point  to  point,  oscillating  around  a 
medium  position,  r^ ,  which  it  approaches  at  infinity. 

This  difference  of  phase,  TW,  towards  which  current  and 
e.m.f.  tend  at  infinity,  is  determined  by  the  expression 


m  (COST.+  /sin  r  J  =   -         , 
M  -h  =  oo 


or,  substituting  for  E  and  7  their  values  from  equations  (23),  and 
since  e~al  =  0,  and  Ateal  (cos  jtt  —  j  sin  pi)  cancels, 

>/~Z       V  - 

m  (cos  r^  +  j  sin  rj  ==  \  —  =  -  = 


hence,  tan  r^  =  —        ~^L  .  (35) 

ag  +  po 

14.  This  angle,  r^  =  0;    that  is,  current  and  e.m.f.    come 
more  and  more  in  phase  with  each  other  when 

ab  —  pg  —  0;  that  is, 
<*  +  P    =  9  +  &,  or 


2ap  2gb 

substituting  (12)  gives 

gr  -  bx  ^  g*  -V 
gx  +  br  ~      2gb 

hence,  expanding,  r  -r-  x  =  g  -?-  6;  (36) 

that  is,  J/ie  ratio  of  resistance  to  inductance  equals  the  ratio  of 
leakage  to  capacity. 


LONG-DISTANCE   TRANSMISSION  LINE  297 

This  angle,  r^  =  45°;   that  is,  current  and  e.m.f.  differ  by 
one-eighth  period  if  +  ab  —  fig  =  ag  +  J3b,  or 


which  gives  rg  +  xb  =  0,  (37) 

which  means  that  two  of  the  four  line-constants,  either  g  and  x 
or  g  and  6,  must  be  zero. 

The  case  where  g  =  0  =  x,  that  is,  a  line  having  only  resistance 
and  distributed  capacity  but  no  self-inductance,  is  approxi- 
mately realized  in  concentric  or  multiple-conductor  cables,  and 
in  these  the  space-phase  angle  tends  towards  45  degrees  lead  for 
infinite  length. 

15.  As  an  example  are  shown  the  characteristic  curves  of  a 
transmission  line  of  the  relative  constants, 

r  :  x  :  g  :  b  =  8  :  32  :  1.25  X  1(T4  :  25  X  10~4   and   e  =  25,000, 
i  =  200  at  the  receiving  circuit,  for  the  conditions 

(a)  Non-inductive  load  in  the  receiving  circuit,  Fig.  80. 

(6)  Wattless  receiving  circuit  of  90  time-degrees  lag,  Fig.  81. 

(c)  Wattless  receiving  circuit  of  90  time-degrees  lead,  Fig.  82. 

These  curves  are  determined  graphically  by  constructing  the 
topographic  circuit  characteristics  in  polar  coordinates  as 
explained  in  "The  Theory  and  Calculation  of  Alternating- 
Current  Phenomena,"  fourth  edition,  Chapter  VII,  paragraphs 
42  to  44,  and  deriving  corresponding  values  of  current,  potential 
difference,  and  phase  angle  therefrom. 

As  seen  from  these  diagrams,  for  wattless  receiving  circuit, 
current  and  e.m.f.  oscillate  in  intensity  inversely  to  each  other, 
with  an  amplitude  of  oscillation  gradually  decreasing  when 
passing  from  the  receiving  circuit  towards  the  generator,  while 
the  space-phase  angle  between  current  and  e.m.f.  oscillates 
between  lag  and  lead  with  decreasing  amplitude.  Approximately 
maxima  and  minima  of  current  coincide  with  minima  and 
maxima  of  e.m.f.  and  zero  phase  angles. 

For  such  graphical  constructions,  polar  coordinate  paper  and 
two  angles  a  and  d  are  desirable,  the  angle  a  being  the  angle 

between  current  and  change  of  e.m.f.,  tan  a  =  -  =  4,  and  the 


298 


TRANSIENT  PHENOMENA 


Distam 


L 


4.2Q 


100 


-80- 


-60, 


-40 


Fig.  80.    Current,  e.m.f.  and  space-phase  angle  between  current  and  e.m.f. 
in  a  transmission  line.     Non-inductive  load. 


480 


•140 


-40 


-80 


ag 


f  C 


\^ 


I 


& 


l/\ 


\ 


Dis  anc 


100 


Fig.  81.     Current,  e.m.f.  and  space-phase  angle  between  current  and  e.m.f. 
in  a  transmission  line.     Inductive  load. 


LONG-DISTANCE  TRANSMISSION  LINE  299 

angle  3  the  angle  between  e.m.f.  and  change  of  current,  tan  d  = 

-  =  20  in  above  instance. 
9 

With  non-inductive  load,  Fig.  80,  these  oscillations  of  intensity 
have  almost  disappeared,  and  only  traces  of  them  are  noticeable 
in  the  fluctuations  of  the  space-phase  angle  and  the  relative 
values  of  current  and  e.m.f.  along  the  line. 

Towards  the  generator  end  of  the  line,  that  is,  towards  rising 
power,  the  curves  can  be  extended  indefinitely,  approaching 
more  and  more  the  conditions  of  non-inductive  circuit.  Towards 
decreasing  power,  however,  all  curves  ultimately  reach  the 
conditions  of  a  wattless  receiving  circuit,  as  Figs.  81  and  82,  at 
the  point  where  the  total  energy  input  into  the  line  has  been 
consumed  therein,  and  at  this  point  the  two  curves  for  lead  and 
for  lag  join  each  other  as  shown  in  Fig.  83,  the  one  being  a 
prolongation  of  the  other,  and  the  power  in  the  line  reverses. 
Thus  in  Fig.  83  energy  flows  from  both  sides  of  the  line  towards 
the  point  of  zero  power  marked  by  0,  where  the  current  and  e.m.f. 
are  in  quadrature  with  each  other,  the  current  being  leading 
with  regard  to  the  power  from  the  left  and  lagging  with  regard 
to  the  power  from  the  right  side  of  the  diagram. 

16.  It  is  of  interest  to  investigate  some  special  cases  of  such 
circuits  of  distributed  constants. 

(A)  Open  circuit  at  the  end  of  the  line. 

Assuming  a  constant  alternating  e.m.f.  El  impressed  upon  a 
circuit  at  one  end  while  the  other  end  of  the  circuit  is  open. 

Counting  the  distance  I  from  the  open  end  of  the  line,  and 
denoting  the  length  of  the  line  by  Z0,  for  I  =  0, 

i  =  /o  =  o, 

and  for  I  =  L, 


hence,  substituting  in  equations  (17), 

0  =  A,  -  A2, 

4i€+*(c°s#o-^  ; 

hence,  A2  =  Al  =  A 


300 


TRANSIENT  PHENOMENA 


Fig.  82.    Current,  e.m.f.  and  space-phase  angle  between  current  and  e.m.f. 
in  a  transmission  line.     Anti-inductive  load. 


En 


x*< 


IH 


t 


:  now  o 


Fig.  83.     Current,  e.m.f.  and  space-phase  angle  between  current  and  e.m.f. 
in  a  transmission  line. 


LONG-DISTANCE  TRANSMISSION  LINE 


301 


and 


e+tU°  (cos/?/,,  -  j  sin/?/0)  +  £-"<«  (cos  /H0  -  j  sin  /?/0) 


hence,  substituting  in  (17), 


T  771 

and 

77?  77T 

.         ~  .     1 


cos       - 


£—')  sin 


cospl0-j  (£+a*°  -  e-^sin 
®pl  -  j  (e+al  -  e-al)smftt 


-  J 


-  ff-^o}  si 


e—  ")  sin  /?Z0 


(38) 


At  Z  =  0,  or  the  open  end  of  the  line,  by  equations  (38), 


and 


2E, 


-«-«•) -sin  ft 


(39) 


The  absolute  values  of  /  and  E  follow  from  equations  (38) 
and  (39)  : 


p  Jy_  v/ 

lV          V 


sn 


T        Tj1      \/   *L 


which  expanded  gives 


and 


and 


;    J  +  2cos2/?/0 


+  2  cos 


(40) 


2  cos  2  /#0 


(41) 


302  TRANSIENT   PHENOMENA 

As  function  of  I,  the  e.m.f.  E  or  the  current  7  is  a  maximum  or 
minimum  for 


£-2al    ±   2  COS  2  /?/     =    Oj 

hence, 


£-2ttZ)  =  ±  2  ft  sin  2  01.  (42) 

For  Z  =  0,  and  since  a  is  a  small  quantity,  the  left  side  of  (42) 
also  is  small,  and  for  values  of  sin  2  fll  approximating  zero,  that 

is,  in  the  neighborhood  of  I  =   --}  or  where  fil  is  a  multiple 

Z  p 

of  a  quadrant,  equation  (42)   becomes  zero.     At  fil  =  2  n-,  or 
the  even  quadrants,  E    is    a    maximum,  /    a    minimum,  at 

01  ~  (2  n  —  1)  -  ,  or  the  odd  quadrants,  E  is  a  minimum,  7  a 
£ 

maximum. 

The  even  quadrants,  therefore,  are  nodes  of  current  and  wave 
crests  of  e.m.f.,  and  the  odd  quadrants  are  nodes  of  e.m.f.  and 
crests  of  current. 

A  maximum  voltage  point,  or  wave  crest,  occurs  at  the  open 
end  of  the  line  at  /  =  0,  and  is  given  by  equation  (41).  As  func- 
tion of  the  length  10  of  the  line  this  is  a  maximum  for 


t~2al*  +  2  cos  2  /?/0    -  0, 

or  a  (e+2al°  -  e-2«*>)  =  2  /?  sin  2  0Z0, 

or  approximately  at 


(43) 


that  is,  when  the  line  is  a  quarter  wave  length  or  an  odd  multiple 
thereof. 


LONG-DISTANCE  TRANSMISSION  LINE  303 


Substituting  in  (41),  /?/0  =  -  gives  (44) 


-f  £-2a<o  _  2 


Since 


for  small  values  of  al0  we  have 


2«l, 

and 

EO=^>  (46) 

which  is  the  maximum  voltage  that  can  occur  at  the  open  end 
of  a  line  with  voltage  El  impressed  upon  it  at  the  other  end. 
Since,  approximately, 

P=  Vxb 

by  (44)  we  have 

(47) 


the  frequency  which  at  the  length  of  line  Z0  produces  maximum 
voltage  at  the  open  end. 

For  the  constants  in  the  example  discussed  in  paragraph  11 
we  have  10  =  200  miles,  r  =  0.21  ohm,  L  =  1.2  X  10~3  henry, 
C  =  0.03   X  1Q-6    farad,   g  =  0,  /  =  208  cycles  per  sec.,  x  = 
1.57  ohms,  z  =--  1.58   ohms,   b  =  39  X  10"6  mho,  a  =  0.53  X 
10-3,  and  #0  =  9.3  Er 

(B)  Line  grounded  at  the  end. 


304  TRANSIENT  PHENOMENA 

17.  Let  the  circuit  be  grounded  or  connected  to  the  return 
conductor  at  one  end,  I  =  0,  and  supplied  by  a  constant  impressed 
e.m.f.  E1  at  the  other  end,  I  =  1Q. 

Then  for  I  =  0, 

771     __      777          f~\ 

and  for  I  =  10, 


hence,  substituting  in  (17), 

0 

and 


hence, 

4i  =  ~~  4z  =  4) 

and 

•*^-  ~x~      i    «? —  7_  \ 


and,  substituting  in  (17), 

(e+al  +  £~al)  cos/?Z  -  j  (e+al-£-al)  sin 


:   *  * 


~a0   cos     0  -  -     £a°  +  £- 
cos  ?Z  -      g+af  +  £-aZ  sin 


(48) 


-  £-0    cos     0  - 
At  the  grounded  end,  I  =  0, 


j  = = (49) 

(s   a^  —  s   a^j  cos  plQ  —  j(^        ~t~£        )  sin  pLQ 

Substituting  (49)  in  (48)  gives 
/  =  J  /   { (fi+°*  +  e-^)  cos  ^Z  -  j  (e+ai  -  e~ai)  sin  pi} 


+tl 


=  \  IQ     -    (e+««  -e-«0  cos  /?Z  -  j  (e 


LONG-DISTANCE  TRANSMISSION  LINE  305 

In  this  case  nodes  of  voltage  and  crests  of  current  appear  at 

I  =  0  and  at  the  even  quadrants,  pl  =  2n  - ,  and  nodes  of  current 

2 

and  crests  of  voltage  appear  at  the  odd  quadrants,  [ft  =  (2n  —  1)  -  • 

& 

(C)  Infinitely  long  conductor. 

18.  If  an  e.m.f.  EQ  is  impressed  upon  an  infinitely  long  con- 
ductor, that  is,  a  conductor  of  such  length  that  of  the  power 
input  no  appreciable  part  reaches  the  end,  we  have,  for  I  =  0, 

E  =  E0 

and  for  I  =  oo, 

E  =  0  and  /  =  0; 


•hence,  substituting  in  (23)  gives 
and 


A2=  0 


hence, 

fy 
I  =  E0  V  77£~al  (cos  pi  +  j  sin  pi) 

(51) 


and 

E  =  E^-*1  (cos  pi  +  j  sin  pi). 

From  (51)  it  follows  that 


7          Y' 

that  is,  an  infinitely  long  conductor  acts  like  an  impedance, 

Z.-V/^-r,-/*,, 

and  the  current  at  every  point  of  the  conductor  thus  has  the  same 
space-phase  angle  to  the  voltage, 

tan  «,  =  -1  - 

T< 


306  TRANSIENT  PHENOMENA 

The  equivalent  impedance  of  the  infinite  conductor  is 

7       a  -  //? 


z, 


YY~g-jb 
.flg-ab 


and  the  space-phase  angle  is 


If  <7  =  0  and  x  =  0,  we  have 


'and 

tan  at  =  1, 

or 

«t  =  45°; 

that  is,  current  and  e.m.f.  differ  by  one-eighth  period. 

This  is  approximately  the  case  in'  cables,  in  which  the  dielectric 
losses  and  the  inductance  are  small. 

An  infinitely  long  conductor  therefore  shows  the  wave  propa- 
gation in  its  simplest  and  most  perspicuous  form,  since  the 
reflected  wave  is  absent. 

(D)  Generator  feeding  into  a  closed  circuit. 

19.  Let  /  =  0  be  the  center  of  the  circuit;  then 

El  =  -  E_t   and  It  =  /_,; 
hence,  E  =  0  at  I  =  0, 

and  the  equations  are  the  same  as  those  of  a  line  grounded  at  the 
end  I  =  0,  which  have  been  discussed  under  (B). 

(E)  Line  of  quarter  wave  length. 

20.  Interesting  is  the  case  of  a  line  of  quarter  wave  length. 
Let  the  length  10  of  the  line  be  one  quarter  wave  of  the  im- 

pressed e.m.f. 

fl.  -  \  •  (54) 


LONG-DISTANCE  TRANSMISSION  LINE 


307 


To  illustrate  the  general  character  of  the  phenomena,  we  may 
as  first  approximation  neglect  the  energy  losses  in  the  circuit, 
that  is,  assume  the  resistance  r  and  the  conductance  g  as  neg- 
ligible compared  with  x  and  6, 

r  =  0  =  g. 


These  values  substituted  in  (14)  give 

a  =  0    and        =  Vxb. 


(55) 


Counting  the  distance  I  from  the  end  of  the  line  10  we  have  for 
=  0, 

EQ  =  e0  +  je0' 


and 


= 


and  at  the  beginning  of  the  line  for  I  —  10, 
and 

and  by  (54)  and  (55), 


L  = 


~2Vxb 
Substituting  (56),  (57),  and  (54)  in  (17)  gives 


(56) 


(57) 


(58) 


and 


or 


B.  -  VI  (4,  +  4.)  =  Vs  (4, 


+ 


and 


^=-/V^(4-42)--yV-(41-42); 


hence,  eliminating  Ax  and  A2  gives  the  relations  between  the 
electric  quantities  at  the  generator  end  of  the  quarter-wave  line, 
Ev  I  v  and  at  the  receiving  end,  Ew  70: 


308 


TRANSIENT  PHENOMENA 


and 


and  the  absolute  values  are 


and 


(59) 


(60) 


which  means  that  if  the  supply  voltage  E1  is  constant,  the  output 
current  70  is  constant  and  lags  90  space-degrees  behind  the 
input  voltage;  if  the  supply  current  II  is  constant,  the  output 
voltage  E0  is  constant,  and  lags  90  space-degrees,  and  inversely. 
A  quarter- wave  line  of  negligible  losses  thus  converts  from 
constant  potential  to  constant  current,  or  from  constant  cur- 
rent to  constant  voltage.  (Constant-potential  constant-current 
transformation.) 
Multiplying  (60)  gives 


or 


E   -- 


hence,  if  70  =  0,  that  is,  the  line  is  open  at  the  end,  E0  =  oo ,  and 
with  a  finite  voltage  supply  to  a  line  of  quarter-wave  length,  an 
infinite  (extremely  high)  voltage  is  produced  at  the  other  end. 

Such  a  circuit  thus  may  be  used  to  produce  very  high  voltages. 

Since  x0  =  I0x  =  total  reactance  and  b0  =  I0b  =  total  sus- 
ceptance  of  the  circuit,  by  (58)  we  have 


(61) 


or  the  condition  of  quarter-wave  length. 


LONG-DISTANCE  TRANSMISSION  LINE 


309 


Substituting  x0  =  2  7r/L0  and  60  =  2  7r/(70,  we  have 

1 


or 


f 


16/2' 
1 


(62) 
(63) 


the  condition  of  quarter-wave  transmission. 

21.  If  the  resistance,  r,  and  the  conductance,  g,  of  a  quarter- 
wave  circuit  are  not  negligible,  substituting  (56),  (54)  and  (57) 
in  (17)  we  have,  for  I  =  0, 


- 


and 


and  for  I  =  L 


and 


From  (64)  it  follows  that 


and 


and  substituting  in  (65)  and  rearranging  we  have 

a*0       \       c~a*0  c+a^0     _     c-a 

T   •  T£ 


(64) 


(65) 


(66) 


and 


(67) 


310  TRANSIENT  PHENOMENA 

or, 


__  £  -  2al0 

and 


(68) 


or,  analogous  to  equation  (59), 


•  °    v  z 


and 


(69) 


In  these  equations  the  second  term  is  usually  small,  due  to 
the  factor  (e+a*°  —  e~a*°),  and  the  first  term  represents  constant 
potential-constant  current  transformation. 

22.  In  a  quarter-wave  line,  at  constant  impressed  e.m.f.  Ev 
the  current  output  1  0  is  approximately  constant  and  lagging  90 
degrees  behind  El  ;  it  falls  off  slightly,  however,  with  increasing 
load,  that  is,  increasing  7  v  due  to  the  second  term  in  equation 
(68);  the  voltage  at  the  end  of  the  line,  E0,  at  constant 
impressed  voltage,  is  approximately  proportional  to  the  load, 
but  does  not  reach  infinity  at  open  circuit,  but  a  finite,  though 
high,  limiting  value. 

Inversely,  at  constant  current  input  the  voltage  output  is 
approximately  constant  and  the  output  current  proportional 
to  the  load. 

The  deviation  from  constancy,  at  constant  Ev  of  70,  or  at 
constant  I  v  of  EQ,  therefore,  is  due  to  the  second  term,  with 
factor  (e+^-  £-'a*°). 

Substituting  (54), 

an 


hence,  al0  is  usually  a  very  small  quantity,  and  e~  ak  =  s  ?2  thus 
can  be  represented  by  the  first  terms  of  the  series  : 


LONG-DISTANCE  TRANSMISSION  LINE  311 

lM2  +  ^LM°+±, 


a:  TT 


hence, 

and 

and,  by  (69), 

and 


2 

£  — 


2 


If  r  and  gi  are  small  compared  with  x  and  6, 

ft  =  VJ  (ay  -  rg  +  xb)  =  \/2> 

and 


a  = 


«/  +  rg  -  xb)  ; 
substituting,  by  the  binomial  theorem, 

fc2)  =  x6  j  [l  +  Q'J 


(70) 


zy  = 


l  + 


gives 


bx/r 


and 


312  TRANSIENT  PHENOMENA 

The  quantity 


a 


may  be  called  the  time  constant  of  the  circuit. 
The  equations  of  quarter-wave  transmission  thus  are 


(72) 
and 


and  the  maximum  voltage  Ea,  at  the  open  end  of  the  circuit,  at 
constant  impressed  e.m.f.  Ev  is 


and  E0  =         ,  (73) 

and  the  current  input  is 


where,  approximately, 


\IZ  -\ 
V  17  ~V  7 


Y      T  c  (75) 


23.  Consider  as  an  example  a  high  potential  coil  of  a  trans- 
former with  one  of  its  terminals  connected  to  a  source  of  high 
potential,  for  testing  its  insulation  to  ground,  while  the  other 
terminal  is  open. 

Assume  the  following  constants  per  unit  length  of  circuit: 
r  =  0.1  ohm,  L  =  0.02  mh.,  C  =  0.01  X  10-6  farad,  and  g  =  0; 
then,  with  a  length  of  circuit  1Q  =  100,  the  quarter-wave  fre- 
quency is,  by  (47), 

/  = -=  =  177  cycles  per  sec., 


LONG-DISTANCE  TRANSMISSION  LINE  313 

or  very  close  to  the  third  harmonic  of  a  60-cycle  impressed 
voltage. 

If,  therefore,  the  testing  frequency  is  low,  59  cycles,  the  circuit 
is  a  quarter  wave  of  the  third  harmonic. 

Assuming  an  impressed  e.m.f.  of  50,000  volts  and  59  cycles, 
containing  a  third  harmonic  of  10  per  cent,  or  El  =  5000  volts  at 
177  cycles,   for  this  harmonic,   we  have  x  =  22.2  ohms  and 
6  -  11.1  X  10-  6  ohm;  hence,  u  =  0.00225 
and 


therefore  at  El  =  5000  volts,  E?  =  1,415,000  volts; 
that  is,  infinity,  as  far  as  insulation  strength  is  concerned. 

Quarter-  wave  circuits  thus  may  be  used,  and  are  used,  to  pro- 
duce extremely  high  voltages,  and  if  a  sufficiently  high  frequency 
is  used  —  100,000  cycles  and  more,  as  in  wireless  telegraphy,  etc. 
-  the  length  of  the  circuit  is  moderate. 

This  method  of  producing  high  voltages  has  the  disadvantage 
that  it  does  not  give  constant  potential,  but  the  high  voltage  is 
due  to  the  tendency  of  the  circuit  to  regulate  for  constant  current, 
which  means  infinite  voltage  at  infinite  resistance  or  open  circuit, 
but  as  soon  as  current  is  taken  off  the  high  potential  point  the 
voltage  falls.  The  great  advantage  of  the  quarter-  wave  method 
of  producing  high  voltage  is  its  simplicity  and  ease  of  insula- 
tion; as  the  voltage  gradually  builds  up  along  the  circuit, 
the  high  voltage  point  or  end  of  circuit  may  be  any  distance 
away  from  the  power  supply,  and  thus  can  easily  be  made 
safe. 

24.  As  a  quarter-wave  circuit  converts  from  constant  poten- 
tial to  constant  current,  it  is  not  possible,  with  constant  voltage 
impressed  upon  a  circuit  of  approximately  a  quarter-wave  length, 
to  get  constant  voltage  at  the  other  or  receiving  end  of  the  circuit. 
Long  before  the  circuit  approaches  quarter-wave  length,  and  as 
soon  as  it  becomes  an  appreciable  part  of  a  quarter  wave,  this 
tendency  to  constant  current  regulation  makes  itself  felt  by  great 
variations  of  voltage  with  changes  of  load  at  the  receiving  end  of 
the  circuit,  constant  voltage  being  impressed  upon  the  generator 
end  ;  that  is,  with  increasing  length  of  transmission  lines  the  volt- 
age regulation  at  the  receiving  end  becomes  seriously  impaired 


314  TRANSIENT  PHENOMENA 

hereby,  even  if  the  line  resistance  is  moderate,  and  the  operation 
of  apparatus  which  require  approximate  constancy  of  voltage 
but  do  not  operate  on  constant  current  —  as  most  synchronous 
apparatus  —  becomes  difficult. 

Hence,  at  the  end  of  very  long  transmission  lines  the  voltage 
regulation  becomes  poor,  and  synchronous  machines  tend  to 
instability  and  have  to  be  provided  with  powerful  steadying 
devices,  giving  induction  motor  features,  and  with  a  line 
approaching  quarter-wave  length,  voltage  regulation  at  the 
receiving  end  ceases. 

In  this  case  the  constant  potential-constant  current  trans- 
formation may  be  used  to  produce  constant  or  approximately 
constant  voltage  at  the  load,  by  supplying  constant  current  to 
the  line;  that  is,  the  transmission  line  is  made  a  quarter- wave 
length  by  modifying  its  constants,  or  choosing  the  proper  fre- 
quency, the  generators  are  designed  to  regulate  for  constant 
current  and  thus  give  a  voltage  varying  with  the  load,  and  are 
connected  in  series  (with  constant  current  generators  series  con- 
nection is  stable,  parallel  connection  unstable)  and  feed  constant 
current,  at  variable  voltage,  into  the  quarter-wave  line.  At 
the  receiving  end  of  the  line,  constant  voltage  then  exists  with 
varying  load,  or  rather  a  voltage,  which  slightly  falls  off  with  the 
load,  due  to  the  power  loss  in  the  line.  To  maintain  constant 
receiver  voltage  at  all  loads,  then,  would  require  a  slight  increase 
of  generator  current  with  increase  of  load,  that  is,  increase  of 
generator  voltage,  which  can  be  produced  by  compounding 
regulated  by  the  voltage. 

In  such  a  quarter-wave  transmission  the  voltage  at  the  receiv- 
ing end  then  remains  constant,  while  the  current  output  from  the 
line  increases  from  nothing  at  no  load.  At  the  generator  end  the 
current  remains  approximately  constant,  increasing  from  no  load 
to  full  load  by  the  amount  required  to  take  care  of  the  line  loss, 
while  the  voltage  at  the  generators  increases  from  nearly  nothing 
at  no  load,  with  increasing  load,  approximately  proportional 
thereto. 

25.  There  is,  however,  a  serious  limitation  imposed  upon 
quarter-wave  transmission  by  considerations  of  voltage;  to  use 
the  transmission  line  economically  the  voltage  throughout  it 
should  not  differ  much,  since  the  insulation  of  the  line  depends 
on  the  maximum,  the  efficiency  of  transmission,  however,  on  the 


LONG-DISTANCE   TRANSMISSION  LINE  315 

average  voltage,  and  a  line  in  which  the  voltage  at  the  two  ends 
is  very  different  is  uneconomical. 

To  use  line  copper  and  line  insulation  economically,  in  a 
quarter-wave  transmission,  the  voltages  at  the  two  ends  should 
be  approximately  equal  at  maximum  load.  These  voltages  are 
related  to  each  other  and  to  the  current  by  the  line  constants,  by 
equations  (72). 

By  these  equations  (neglecting  the  term  with  u),  reduced  to 
absolute  values,  we  have  approximately 


-V?, 


and 

and  if  el  =  e0, 

hence,  the  power  is 


=  V 


or 

e*  =  Po  V -  ;  (76) 

y 

hence,  the  voltage  e0  required  to  transmit  the  power  pQ  without 
great  potential  differences  in  the  line  depends  on  the  power  p0 
and  the  line  constants,  and  inversely. 

26.  As  an  example  of  a  quarter- wave  transmission  may  be 
considered  the  transmission  of  60,000  kilowatts  over  a  distance 
of  700  miles,  for  the  supply  of  a  general  three-phase  distribution 
system,  of  95  per  cent  power  factor,  lag. 

The  design  of  the  transmission  line  is  based  on  a  compromise 
between  different  and  conflicting  requirements:  economy  in 
first  cost  requires  the  highest  possible  voltage  and  smallest  con- 
ductor section,  or  high  power  loss  in  the  line ;  economy  of  opera- 
tion requires  high  voltage  and  large  conductor  section,  or  low 
power  loss;  reliability  of  operation  of  the  line  requires  lowest 


316  TRANSIENT  PHENOMENA 

permissible  voltage  and  therefore  large  conductor  section  or 
high  power  loss;  reliability  of  operation  of  the  receiving  system 
requires  good  voltage  regulation  and  thus  low  line  resistance, 
etc.,  etc. 

Assume  that  the  maximum  effective  voltage  between  the  line 
conductors  is  limited  to  120,000,  and  that  there  are  two  sepa- 
rate pole  •  lines,  each  carrying  three  wires  of  500,000  circular 
mils  cross  section,  placed  6  feet  between  wires,  and  provided 
with  a  grounded  neutral. 

If  there  were  no  energy  losses  in  the  line  and  no  increase  of 
capacity  due  to  insulators,  etc.,  the  speed  of  propagation  would 
be  the  velocity  of  light,  S  =  188,000  miles  per  second,  and  the 
quarter-wave  frequency  of  a  line  of  10  =  700  miles  would  be 

S 
/  =  —  =67  cycles  per  sec. ; 

4     LQ 

hence,  fairly  close  to  the  standard  frequency  of  60  cycles. 

The  loss  of  power  in  the  line,  and  thus  the  increase  of  induc- 
tance by  the  magnetic  field  inside  of  the  conductor  (which  would 
not  exist  in  a  conductor  of  perfect  conductivity  or  zero  resistance 
loss),  the  increase  of  capacity  by  insulators,  poles,  etc.,  lowers  the 
frequency  below  that  corresponding  to  the  velocity  of  light  and 
brings  it  nearer  to  60  cycles. 

In  a  line  as  above  assumed  the  constants  per  mile  of  double 
conductor  are:  r  =  0.055  ohm;  L  =  0.001  henry,  and  C  = 
0.032  X  10~6  farad,  and,  neglecting  the  conductance,  g  =  0,  the 
quarter-wave  frequency  is 

/  = — =  =  63  cycles  per  sec. 

Either  then  the  frequency  of  63  cycles  per  second,  or  slightly 
above  standard,  may  be  chosen,  or  the  line  inductance  or  line 
capacity  increased,  to  bring  the  frequency  down  to  60  cycles. 

Assuming  the  inductance  increased  to  L  =  0.0011  henry 
gives  /  =  =  60  cycles  per  second,  and  the  line  constants  then  are 
£0  =  700  miles;  /  =  60  cycles  per  second;  r  =  0.055  ohm;  L  = 
0.0011  henry;  C  =  0.032  X  10~6  farad,  and  g  =  0;  hence, 
x  =  0.415  ohm;  z  =  0.42  ohm;  Z  =  0.055  -  0.415  j  ohm; 


LONG-DISTANCE   TRANSMISSION  LINE  317 

6  =  12.1  X  10"6  mho;  y  =  12.1  X  10~6  mho,  and  Y  =  -  j  12.1 
X  10~6  mho,  and 


v/ 


-  =  186, 

y 


0.066, 


p  =  2.247  X  10-3 

a  =  up  =  0.148  X  10-3. 

At  60,000  kilowatts  total  input,  or  20,000  kilowatts  per  line, 

120  000 
and  120,000  volts  between  lines,  or  -  '-1—  =  69,000  volts  per 

line,  and  about  95  per  cent  power  factor,  the  current  input  at 
full  load  is  306  amp.  per  line  (of  two  conductors  in  multiple). 

To  get  at  full  load  p  =  20  X  106  watts,  approximately  the 
same  voltage  at  both  ends  of  the  line,  by  equation  (67),  we  must 
have 


y 

or  e  =  61,000  volts. 

Assuming  therefore  at  the  receiving  end  the  voltage  of  110,000 
between  the  lines,  or,  63,500  volts  per  line,  and  choosing  the 
output  current  as  zero  vector,  and  counting  the  distance  from 
the  receiving  end  towards  the  generator,  we  have  for  /  =  0, 

.        ~     .  0  0> 


and  the  voltage,  at  95  per  cent  power  factor,  or  Vl  —  0.952 
=  0.312  inductance  factor,  is 

E  =  E0  =  e0  (0.95  -  0.312  j) 
=  60,300  -  19,800  j. 

Substituting  these  values  in  equations  (72)  gives 

.        \JE1  -  0.104  (60,300  -  19,800  j)} 
*°  =  186  +  12  j 

60,300  -  1 9,800  /  =  (186  +  12  j)  {j  I,  -  0.104  i0}; 


318 


TRANSIENT  PHENOMENA 


hence,      JE1  =   (186  +  12  j)  iQ  +  (6250  -  2060  j) 
and          .(^  60,300 -19,800f+  Q  ^  ^ 

=  317  -  128  j  +  0.104  i0, 
and  the  absolute  values  are 


and 


e,  =  V(186  i0  +  6250)2  +  (12  t0  -  2060)' 
\  =  V(317  +  0.104  i, 


1282 


70  L^ 


Cos. 


0-1— 


80  32 


70  28 


500  50  20 


400  40  16 


300  30  12 


100  10  4 


6  10  14  18 

Power  Output  per  Phase  Po 
.Megawatts 


26 


Fig.  84.    Long-distance  quarter-wave  transmission. 

herefrom  the  power  output  and  input,  efficiency,  power  factor, 
etc.,  can  be  obtained. 

In  Fig.  84,  with  the  power  output  per  phase  as  abscissas,  are 
shown  the  following  quantities  :  voltage  input  e1  and  output  e0, 


LONG-DISTANCE  TRANSMISSION  LINE  319 

in  drawn  lines;  amperes  input  t\  and  output  t0,  in  dotted  lines; 
power  input  pl  and  output  p0,  in  dash-dotted  lines,  and  efficiency 
and  power  factor  in  dashed  lines. 

As  seen,  the  power  factor  at  the  generator  is  above  93  per  cent 
leading,  and  the  efficiency  reaches  nearly  85  per  cent. 

At  full  load  input  of  20,000  kilowatts  per  phase,  and  95  per 
cent  power  factor,  lagging,  of  the  output,  the  generator  voltage 
is  58,500,  or  still  8  per  cent  below  the  output  voltage  of  63,500. 
The  generator  voltage  equals  the  output  voltage  at  10  per  cent 
overload,  and  exceeds  it  by  14  per  cent  at  25  per  cent  overload. 

To  maintain  constant  voltage  at  the  output  side  of  the  line, 
the  generator  current  has  to  be  increased  from  342  amperes  at 
no  load  to  370  amperes  at  full  load,  or  by  8.2  per  cent,  and 
inversely,  at  constant-current  input,  the  output  voltage  would 
drop  off,  from  no  load  to  full  load,  by  about  8  per  cent.  This, 
with  a  line  of  15  per  cent  resistance  drop,  is  a  far  closer  voltage 
regulation  than  can  be  produced  by  constant  potential  supply, 
except  by  the  use  of  synchronous  machines  for  phase  control. 


CHAPTER  III. 

THE    NATURAL    PERIOD    OF    THE    TRANSMISSION    LINE. 

27.  An  interesting  application  of  the  equations  of  the  long 
distance  transmission  line  given  in  the  preceding  chapter  can  be 
made  to  the  determination  of  the  natural  period  of  a  transmis- 
sion line;  that  is,  the  frequency  at  which  such  a  line  discharges 
an  accumulated  charge  of  atmospheric  electricity  (lightning),  or 
oscillates  because  of  a  sudden  change  of  load,  as  a  break  of  circuit, 
or  in  general  a  change  of  circuit  conditions,  as  closing  the  circuit, 
etc. 

The  discharge  of  a  condenser  through  a  circuit  containing  self- 
inductance  and  resistance  is  oscillating  (provided  the  resistance 
does  not  exceed  a  certain  critical  value  depending  upon  the 
capacity  and  the  self-inductance)  ;  that  is,  the  discharge  current 
alternates  with  constantly  decreasing  intensity.  The  frequency 
of  this  oscillating  discharge  depends  upon  the  capacity  C  and 
the  self  -inductance  L  of  the  circuit,  and  to  a  much  lesser  extent 
upon  the  resistance,  so  that,  if  the  resistance  of  the  circuit  is  not 
excessive,  the  frequency  of  oscillation  can,  by  neglecting  the 
resistance,  be  expressed  with  fair,  or  even  close,  approximation 
by  the  formula 


An  electric  transmission  line  represents  a  circuit  having 
capacity  as  well  as  self-inductance  ;  and  thus  when  charged  to  a 
certain  potential,  for  instance,  by  atmospheric  electricity,  as  by 
induction  from  a  thunder-cloud  passing  over  or  near  the  line, 
the  transmission  line  discharges  by  an  oscillating  current. 

Such  a  transmission  line  differs,  however,  from  an  ordinary 
condenser  in  that  with  the  former  the  capacity  and  the  self- 
inductance  are  distributed  along  the  circuit. 

In  determining  the  frequency  of  the  oscillating  discharge  of 
such  a  transmission  line,  a  sufficiently  close  approximation  is 

320 


NATURAL  PERIOD  OF  TRANSMISSION  LINE  321 

obtained  by  neglecting  the  resistance  of  the  line,  which,  at  the 
relatively  high  frequency  of  oscillating  discharges,  is  small  com- 
pared with  the  reactance.  This  assumption  means  that  the 
dying  out  of  the  discharge  current  through  the  influence  of  the 
resistance  of  the  circuit  is  neglected,  and  the  current  assumed 
as  an  alternating  current  of  approximately  the  same  frequency 
and  the  same  intensity  as  the  initial  waves  of  the  oscillating 
discharge  current.  By  this  means  the  problem  is  essentially 
simplified. 

28.  Let  10  =  total  length  of  a  transmission  line;  I  =  the  dis- 
tance from  the  beginning  of  the  line;  r  =  resistance  per  unit 
length;  x  =  reactance  per  unit  length  =  2  nfL,  where  L  = 
inductance  per  unit  length;  g  =  conductance  from  line  to  return 
(leakage  and  discharge  into  the  air)  per  unit  length;  b  =  capacity 
susceptance  per  unit  length  =  2  nfC,  where  C  =  capacity  per 
unit  length. 

Neglecting  the  line  resistance  and  line  conductance, 

r  =  0  and  g  =  0, 

the  line  constants  a  and  /?,  by  equations  (14),  Chapter  II,  then 
assume  the  form 

a  =  0  and  ft  =  Vxb,  (1) 

and  the  line  equations  (17)  of  Chapter  II  become 
/  =  (AA  -  A2)  cos  pi  -  j  (Aj  +  A,)  sin  pi 
and 


E  =  V  ^    (A,  +  A2)cos  fl  -  /  (4i  -  42)  sin  pi 

or  writing 

4i  ~  42  =  <7i  and  4i  +  42  =  Q» 

and  substituting 


c 

we  have 

and  •        • l  *  • 2 


(3) 


322  TRANSIENT  PHENOMENA 

A  free  oscillation  of  a  circuit  implies  that  energy  is  neither 
supplied  to  the  circuit  nor  abstracted  from  it.  This  means  that 
at  both  ends  of  the  circuit,  I  =  0  and  I  =  1Q,  the  power  equals  zero. 

If  this  is  the  case,  the  following  conditions  may  exist: 

(1)  The  current  is  zero  at  one  end,  the  voltage  zero  at  the 
other  end. 

(2)  Either  the  current  is  zero  at  both  ends  or  the  voltage  is 
zero  at  both  ends. 

(3)  The  circuit  has  no  end  but  is  closed  upon  itself. 

(4)  The  current  is  in  quadrature  with  the  voltage.     This  case 
does  not  represent  a  free  oscillation,  since  the  frequency  depends 
also  on  the  connected  circuit,  but  rather  represents  a  line  supply- 
ing a  wattless  or  reactive  load. 

In  free  oscillation  the  circuit  thus  must  be  either  open  or 
grounded  at  its  ends  or  closed  upon  itself. 

(1)  Circuit  open  at  one  end,  grounded  at  other  end. 

29.  Assuming  the  circuit  grounded  at  I  =  0,  open  at  /  =  Z0, 
we  have  for  I  =  0, 

#  =  #o  =  0, 

and  for  I  =  Z0, 

!  -I.i-D; 

hence,  substituting  in  equations  (3),  at  I  —  0, 


hence, 

7  =  C,  cos  81 
and 


(4) 


and  at  I  =  Z0, 


!  cos  /?Z0  =  0, 


and  since  Ct  cannot  be  zero  without  the  oscillation  disappearing 
altogether, 

cos#0  =  0;  (5) 

hence, 

#0  =  (2n-l),  (6) 


NATURAL  PERIOD  OF   TRANSMISSION  LINE  323 

where  n  =  1,  2,  3  ...  or  any  integer  and 

£11  =  (2n-  l)~l.  (7) 

Z  L0 

Substituting  (1)  in  (6)  gives 

ft-V*;-V^,  (8) 

z  Lo 

or  substituting  for  x  and  b,  x  =  2  Tr/L  and  6  =  2  7r/<7,  gives 


or 


is  the  frequency  of  oscillation  of  the  circuit. 

The  lowest  frequency  or  fundamental  frequency  of  oscillation 
is,  for  n  =  1, 


and  besides  this  fundamental  frequency,  all  its  odd  multiples  or 
higher  harmonics  may  exist  in  the  oscillation 

f**  (2n-l)/r  (11) 

Writing  L0  =  Z0L  =  total  inductance,  and  C0  =  10C  =  total 
capacity  of  the  circuit,  equation  (9)  assumes  the  form 

(12) 


The  fundamental  frequency  of  oscillation  of  a  transmission 
line  open  at  one  end  and  grounded  at  the  other,  and  having  a 
total  inductance  L0  and  a  total  capacity  (70,  is,  neglecting  energy 
losses, 

fl    =    ~         rr-TT  ' 


324 


TRANSIENT  PHENOMENA 


while  the  frequency  of  oscillation  of  a  localized  inductance  L0 
and  localized  capacity  (70,  that  is,  the  frequency  of  discharge  of 
a  condenser  CQ  through  an  inductance  L0,  is 

/  =  ^=  •  d3) 

The  difference  is  due  to  the  distributed  character  of  L0  and  C0 
in  the  transmission  line  and  the  resultant  phase  displacement 
between  the  elements  of  the  line,  which  causes  the  inductance 
and  capacity  of  the  line  elements,  in  their  effect  on  the  frequency, 
not  to  add  but  to  combine  to  a  resultant,  which  is  the  projection 

2 

of  the  elements  of  a  quadrant,  on  the  diameter,  or  -  times  the 

n 

sum,  just  as,  for  instance,  the  resultant  m.m.f.  of  a  distributed 

2 
armature  winding  of  n  turns  of  i  amperes  is  not  ni  but  -  ni. 

7t 

Hence,  the  effective  inductance  of  a  transmission  line  in  free 
oscillation  is 

L>       2IL 

L0   •-     -  L0L 

n 


and  the  effective  capacity  is 


(14) 


and  using  the  effective  values  L0'  and  C0',  the  fundamental 
frequency,  equation  (11),  then  appears  in  the  form 


2*  ^fLJc^ ' 

that  is,  the  same  value  as  found  for  the  condenser  discharge. 

In  comparing  with  localized  inductances  and  capacities,  the 
distributed  capacity  and  inductance,  in  free  oscillation,  thus  are 
represented  by  their  effective  values  (13)  and  (14). 

30.  Substituting  in  equations  (4), 

Cl  =  <>i  +  jcv  (16) 

gives 

I  =  (cl  +  jc2)  cos  ftl 


and 


(17) 


NATURAL  PERIOD  OF  TRANSMISSION  LINE 


325 


By  the  definition  of  the  complex  quantity  as  vector  represen- 
tation of  an  alternating  wave  the  cosine  component  of  the  wave 
is  represented  by  the  real,  the  sine  component  by  the  imaginary 
term;  that  is,  a  wave  of  the  form  ct  cos  2  nft  +  c2sin  2  nft  is 
represented  by  cl  +  jc2J  and  inversely,  the  equations  (17),  in 
their  analytic  expression,  are 

i  =  (ct  cos  2  nft  +  c2  sin  2  xft)  cos  ftl 


and 


V/Z 
e  -\n 


(c2  cos  2  nft  -  c1  sin  2  rft]  sin  pi. 


Substituting   (7)  and  (11)  in  (18),  and  writing 
0  =  2  rfj,  and  r  =  ^~ 

2  *0 

gives 

i  ==  {cjcos  (2n-  1)6  +  c2  sin  (2n-  1)0}  cos  (2n  -I)  r 

=  ccos  (2n  -  1)  (0  -  r)  cos  (2  n  -  l)r 
and 


(18) 


(19) 


e  =•  Vc2  cos  (2  w-l)^-c1 


-  \  -  c  sin  (2  n  -  1)  (0  -  r)  sin  (2  n  -  l)r, 


(20) 


where 


tan  (2  n  -  1)  r  =  J      and     c  = 


(21) 


In  the  denotation  (19),  0  represents  the  time  angle,  with  the 
complete  cycle  of  the  fundamental  frequency  of  oscillation  as 
one  revolution  or  360  degrees,  and  r  represents  the  distance 
angle,  with  the  length  of  the  line  as  a  quadrant  or  90  degrees. 
That  is,  distances  are  represented  by  angles,  and  the  whole  line 
is  a  quarter  wave  of  the  fundamental  frequency  of  oscillation. 
This  form  of  free  oscillation  may  be  called  quarter-wave  oscillation. 

The  fundamental  or  lowest  discharge  wave  or  oscillation  of 
the  circuit  then  is 

il  =  c  cos  (0  —  fj)  cos  T 


and 


el  =      -  \  —  c  sin  (0  — 


sn  r. 


(22) 


326 


TRANSIENT  PHENOMENA 


With  this  wave  the  voltage  is  a  maximum  at  the  open  end  of 
the  line,  I  =  Z0,  and  gradually  decreases  to  zero  at  the  other  end 
or  beginning  of  the  line,  I  =  0. 

The  current  is  zero  at  the  open  end  of  the  line,  and  gradually 
increases  to  a  maximum  at  I  =  0,  or  the  grounded  end  of  the 
line. 

Thus  the  relative  intensities  of  current  and  potential  along 
the  line  are  as  represented  by  Fig.  85,  where  the  current  is  shown 
as  7,  the  voltage  as  E. 


Fig.  85.    Discharge  of  current  and  e.in.f.  along  a  transmission  line  open  at 
one  end.     Fundamental  discharge  frequency. 


The  next  higher  discharge  frequency,  for  n  =  2,  gives 
i3  =  c3  cos  3  (0  —  7-3)  cos  3  r 


and 


=  -  c3  V  - 


sin  3  (0  -  r3)  sin  3  T. 


(23) 


Here  the  voltage  is  again  a  maximum  at  the  open  end  of  the 
line,  I  =  10,  or  r  =  -  =  90°,  and  gradually  decreases,  but  reaches 


zero  at  two-thirds  of  the  line,   I  = 


2L 


or 


-=60°,  then 

o 


increases  again  in  the  opposite  direction,  reaches  a  second  but 

opposite  maximum  at  one-third  of  the  line,  I  =  -| ,  or  r  =     =  30°, 

o  o 

and  decreases  to  zero  at  the  beginning  of  the  line.  There  is  thus 
a  node  of  voltage  at  a  point  situated  at  a  distance  of  two-thirds 
of  the  length  of  the  line. 

The  current  is  zero  at  the  end  of  the  line,  /  =  Z0,  rises  to  a 
maximum  at  a  distance  of  two-thirds  of  the  length  of  the  line, 
decreases  to  zero  at  a  distance  of  one-third  of  the  length  of  the 
line,  and  rises  again  to  a  second  but  opposite  maximum  at  the 


NATURAL  PERIOD  OF  TRANSMISSION  LINE 


327 


beginning  of  the  line,  I  =  0.    The  current  thus  has  a  node  at  a 
point  situated  at  a  distance  of  one-third  of  the  length  of  the  line. 


Fig.  86.     Discharge  of  current  and  e.m.f.  along  a  transmission 
line  open  at  one  end. 

The  discharge  waves,  n  =  2,  are  shown  in  Fig.  86,  those  with 
n  =  3,  with  two  nodal  points,  in  Fig.  87. 


\ 


A 


/ 


Fig.  87.    Discharge  of  current  and  e.m.f.  along  a  transmission 
line  open  at  one  end. 

31.  In  case  of  a  lightning  discharge  the  capacity  C0  is  the 
capacity  of  the  line  against  ground,  and  thus  has  no  direct 
relation  to  the  capacity  of  the  line  conductor  against  its  return. 
The  same  applies  to  the  inductance  L0. 

If  d  =  diameter  of  line  conductor,  lh=  height  of  conductor 
above  ground,  and  10  =  length  of  conductor,  the  capacity  is 

1.11  X  10-8Z 


?;- 


4L 


. 
°,mmf. 


the  self-inductance  is 


inmh. 


(24) 


328  TRANSIENT  PHENOMENA 

.  The  fundamental  frequency  of  oscillation,  by  substituting  (24) 
in  (10),  is 

1  7.5  X  IP. 

'         '  (25) 


that  is,  the  frequency  of  oscillation  of  a  line  discharging  to  ground 
is  independent  of  the  size  of  line  wire  and  its  distance  from  the 
ground,  and  merely  depends  upon  the  length,  /0,  of  the  line,  being 
inversely  proportional  thereto. 

We  thus  get  the  numerical  values, 
Length  of  line 

100  miles 
16  X  106  cm. 

hence  frequency, 

£  =       4700  2350  1570  1175  940  783    587    470  cycles  per  sec. 

As  seen,  these  frequencies  are  comparatively  low,  and  especially 
with  very  long  lines  almost  approach  alternator  frequencies. 

The  higher  harmonics  of  the  oscillation  are  the  odd  multiples 
of  these  frequencies. 

Obviously  all  these  waves  of  different  frequencies  represented 
in  equation  (20)  can  occur  simultaneously  in  the  oscillating  dis- 
charge of  a  transmission  line,  and,  in  general,  the  oscillating 
discharge  of  a  transmission  line  is  thus  of  the  form 


n  cn  cos  (2  n  —  1)  (0  —  pn)  cos  (2  n  —  1)  T, 


=-\/iix 


e  =  ~  VS  2>cn  sin  (2  n  -  1)  (6  -  rj  sin  (2  n  -  1)  r. 


(26) 


A  simple  harmonic  oscillation  as  a  line  discharge  would  require 
a  sinoidal  distribution  of  potential  on  the  transmission  line  at  the 
instant  of  discharge,  which  is  not  probable,  so  that  probably  all 
lightning  discharges  of  transmission  lines  or  oscillations  produced 
by  sudden  changes  of  circuit  conditions  are  complex  waves  of 
many  harmonics,  which  in  their  relative  magnitude  depend  upon 
the  initial  charge  and  its  distribution  —  that  is,  in  the  case  of  the 
lightning  discharge,  upon  the  atmospheric  electrostatic  field  of 
force. 


NATURAL  PERIOD  OF  TRANSMISSION  LINE  329 

The  fundamental  frequency  of  the  oscillating  discharge  of  a 
transmission  line  is  relatively  low,  and  of  not  much  higher  mag- 
nitude than  frequencies  in  commercial  use  in  alternating-current 
circuits.  Obviously,  the  more  nearly  sinoidal  the  distribution 
of  potential  before  the  discharge,  tfye  more  the  low  harmonics 
predominate,  while  a  very  unequal  distribution  of  potential, 
that  is  a  very  rapid  change  along  the  line,  causes  the  higher  har- 
monics to  predominate. 

32.  As  an  example  the  discharge  of  a  transmission  line  may  be 
investigated,  the  line  having  the  following  constants  per  mile : 
r  -  0.21  ohm;  L  =  1.2  X  10~3  henry;  C  =  0.03  X  10~6  farad, 
and  of  the  length  Z0  =  200;  hence,  by  equations  (10),  (19), 
/!  =  208  cycles  per  sec.;  0  =  1315  t,  and  T  =  0.00785  Z,  when 
charged  to  a  uniform  voltage  of  e0  =  60,000  volts  but  with  no 
current  in  the  line  before  the  discharge,  and  the  line  then 
grounded  at  one  end,  Z  =  0,  while  open  at  the  other  end,  Z  =  Z0. 

Then,  for  t  =  0  or  6  =  0,  i  =  0  for  all  values  of  r  except  T  =  0; 
hence,  by  (26), 

cos  (2  n  —  1)  -jTn  =  0, 

and  thus 

(2»=-l)y.-?  (27) 

and 

cos  (2  n  -  1)  (e  -  rn)  =  sin  (2  n  -  1)  0, 
sin  (2  n  -  1)  (0  -  Tn)  =  -  cos  (2  n  -  1)  0; 

hence, 

i  =  lL,n  cn  sin  (2  n  -  1)  0  cos  (2  n  -  1)  T 


and 


e  =  \  -  ^ncn  cos  (2  n  -  1)  6  sin  (2  n  -  1)  r. 


(28) 


Also  for  t  =  0,  or  0  =  0,  e  =  e0  for  all  values  of  T  except  T  =  0; 
hence,  by  (28), 

"c.ffln  <2  » -1)'1-  .(29) 


330  TRANSIENT  PHENOMENA 

From  equation  (29),  the  coefficients  cn  are  determined  in  the 
usual  manner  of  evaluating  a  Fourier  series,  that  is,  by  multiply- 
ing with  sin  (2  m  —  1)  r  (or  cos  (2m  -  1)  r)  and  integrating: 

ren  sin  (2  m  —  1)  r  dr  = 
• 

\^2,ncn      I     sin  (2n  -  l)rsin  (2m-  1)  r  dr. 

C       j  JQ 

Since 

J    sin  (2  n  —  1)  r  sin  (2  m  —  1)  r  dr 

rcos  2  (n  -  m)  r  -  cos  2  (n  +  m  —  1)  r 
J  —  -  dT' 

which  is  zero  forn  =  ±  m,  while  for  m  =  n  the  term 

X17  cos  2  (rz.  -  m)  r   7         T71"  dr      n 
—       -*-J.    ^  =  2 

and 

cos  (2  n  -  1)  Tl*  2  e0 


we  have 


and 

c    =  -     4  g°     -  V/~:  (30) 

(2n  -  !)^VL? 

hence, 

4      .  /C  A    sin  (2  n  -  1)  0  cos  (2  TI  -  1)  r 

*  - 


C  (  .  sin  3  6  cos  3  T     sin  5  6  cos  5  r 


(31) 

_  (  .                     sin  3  6  cos  3  r      sin  5  6  cos  5  r  ) 

=  382  <  sin  0  cos  r  H —       -  +  -       — —    —  +  •  •  •  > , 

in  amperes, 


NATURAL  PERIOD  OF  TRANSMISSION  LINE  331 

and 

4      A    cos  (2  n  -  1)  0  sin  (2  n  -  1)  r 
TT  °   i  2/1  —  1 

cos  3  #  sin  3  r      cos  5  #  sin  5  r  > 

cos  0  sm  T  -| 1 — h  •  •  •  J 

o  5  ) 

(32) 

_  (  cos  3  6  sin  3  r     cos  5  0  sin  5  r  ) 

=  76,400  ]  cos  0  sm  r  +  -  —  + [-•••{» 

o  5  ) 

in  volts. 

33.  As  further  example,  assume  now  that  this  line  is  short- 
circuited  at  one  end,  I  =  0,  while  supplied  with  25-cycle  alter- 
nating power  at  the  other  end,  I  =  /0,  and  that  the  generator 
voltage  drops,  by  the  short  circuit,  to  30,000,  and  then  the  line 
cuts  off  from  the  generating  system  at  about  the  maximum  value 
of  the  short-circuit  current,  that  is,  at  the  moment  of  zero  value 
of  the  impressed  e.m.f. 

At  a  frequency  of /0  =  25  cycles,  the  reactance  per  unit  length 
of  line  or  per  mile  is 

x  =  2  TT/OL  =  0.188  ohm 
and  the  impedance  is 

z  =  Vr2  +  x*  =  0.283  ohm, 
or,  for  the  total  line, 

z0  =  I0z  =  56.6  ohms; 

hence,  the  approximate  short-circuit  current 
e      30,000 


and  its  maximum  value  is 

i0  -  530  X  \/2  =  750  amp. 

Therefore,  in  equations   (26),  at  time  t  =  0,  or  0  =  0,  e=  0 
for  all  values  of  T  except  T  =  — ;  hence, 

Zi 

sin  (2  n  -  1)  yn  =  0, 
or,  yn  -  0, 


332  TRANSIENT  PHENOMENA 

and  thus 

i  =  ^n  cn  cos  (2  n.  —  1)  6  cos  (2  n  —  1)  r 
and  (33) 

e=    -y5]£ncnsin(2n  -  1)#  sin  (2  ri  -  l)r. 

However,  at  t  =  0,  or  6  =  0,  for  all  values  of  r  except  T  =  ^,    ' 


hence,  substituting  in  (33), 

A 
t'0  -  2/n  cn  cos  (2  n  -  1)  r. 


(34) 


From  equation  (34),  the  coefficients  cn  are  determined  in  the 
same  manner  as  in  the  preceding  example,  by  multiplying  with 
cos  (2  n  —  1)  r  and  integrating,  as 


hence, 
i  =  — 


(2n-l)*' 

cos  (2n  -  l)0cos(2n  -  1)  r 
2n  -  1 


(35) 


4  i'   (  cos  3  6  cos  3  r      cos  5  0  cos  5  r 

=  — -    cos  0  cos  T — - — 


(36) 


(                        cos  3  6  cos  3  T      cos  5  0  cos  5  r 
956 )  cos  0  cos  T +  - 


in  amperes, 


and 


4i 


n  -  1)  0  sin  (2  n  -  1)  T 


4  t*     [L  (  .  sin  3  /9  sin  3  T     si 

-V-  Utfmnr-         -^      -  +  - 


2n  -  1 

sin  5  (9  sin  5  T 


-  + 


(37) 


(  ,                  sin  3  6  sin  3  r    sin  5  6  sin  5  T  ) 

=  191,200 )  sin  6  sin  T—      — — -     -  + — +•••{ 


in  volts. 


NATURAL  PERIOD  OF  TRANSMISSION  LINE  333 


The  maximum  voltage  is  reached  at  time  6  =  - ,  and  is 


e  = 


and  since  the  series 


4  i     IL      .  sin  3  r      sin  5  r 


sin  3  T      sin  5  r 
sm  T  +  - 


the  maximum  voltage  is 

e  =  i  V/^  =  300,000  volts. 


As  seen,  very  high  voltages  may  be  produced  by  the  interrup- 
tion of  the  short-circuit  current. 

(2a)  Circuit  grounded  at  both  ends. 

34.  The  method  of  investigation  is  the  same  as  in  paragraph 
29;  the  terminal  conditions  are,  for  I  =  0, 


E=  0, 


and  for  I  =  I, 


Substituting  Z  =  0  into  equations  (3)  gives 


hence, 


/  =  C,  cos  /M, 


sn 


Substituting  Z  =  Z0  in  (38)  gives 

&  -  o  -  -  jev 

hence, 

sin  £HQ  =  0,  or  /?£  =  TITT, 


(38) 


(39) 


334  TRANSIENT  PHENOMENA 

and,  in  the  same  manner  as  in  (1), 

pi  =  n^l  =  nr; 


(40) 


that  is,  the  length  of  the  line,  Z0,  represents  one  half  wave,  or 
r  =  TT,  or  a  multiple  thereof. 


n 


210VLC      2VL0C0 


and  the  fundamental  frequency  of  oscillation  is 

1 


210VLC 


and 


(41) 

(42) 
(43) 


that  is,  the  line  can  oscillate  at  a  fundamental  frequency  fv  for 
which  the  length,  1OJ  of  the  line  is  a  half  wave,  and  at  all  multiples 
or  higher  harmonics  thereof,  the  even  ones  as  well  as  the  odd  ones. 

This  kind  of  oscillation  may  be  called  a  half-wave  oscillation. 

35.  Unlike  the  quarter-wave  oscillation,  which  contains  only 
the  odd  higher  harmonics  of  the  fundamental  wave,  the  half- 
wave  oscillation  also  contains  the  even  harmonics  of  the  funda- 
mental frequency  of  oscillation. 

Substituting  Ot  =  cl  +  j'c2  into  (38)  gives 


and 


jc2)  cos 


(44) 


and  replacing  the  complex  imaginary  by  the  analytic  expression, 
that  is,  the  real  term  by  cos  2  nft,  the  imaginary  term  by  sin  2  nft, 
gives 

i  —  { cx  cos  2  nft  +  c2  sin  2  nft]  cos  pi 
and 


e 


-s/i 


-  {c2  cos  2  nft  -  cl  sin  2 


sin  pl} 


NATURAL  PERIOD  OF   TRANSMISSION  LINE  335 

and  substituting 

we  have  (45) 

2  nft  =  nO] 

then  (44)  gives,  by  (40) : 

i  =  (c1  cos  nO  +  c2  sin  nO)  cos  nr 
and 


or  writing 
and 


e  =  \  -  (ca  cos  7i#  —  ct  sin  nd)  sin  nr; 
*  C 

ct  =  c  cos  nj    \ 
c2  =  c  sin  n/-    J 

i  =  c  cos  n  (0  —  r)  cos  nr 


(46) 


(47) 


-  c  y  —  sin  n  (0  —  7-)  s 


(48) 


gives 

and 

e  =  —  c  V  —  sin  n  (U  —  r)  sin  nr, 

and  herefrom  the  general  equations  of  this  half-wave  oscillation  are 

co 

t  =  7,7ic»  cos  n  (0  —  r«)  cos 


and 


e  =  —  y  —  X  »  cn  sin  n  (0  —  fn)  sin 


(49) 


(26)  Circuit  open  at  both  ends. 
36.   For  Z  =  0  we  have 


hence, 

and 

and 


0 


=  -        sn 


(50) 


while  for  I  =  L,   7=0; 


336 

hence, 


TRANSIENT  PHENOMENA 


sin  /?Z0  =  0,  or  /?Z 


(51) 


that  is,  the  circuit  performs  a   half-wave  oscillation  of  funda- 
mental frequency, 

1 

(52) 


and  all  its  higher  harmonics,  the  even  ones  as  well  as  the  odd  ones 
have  a  frequency 

/  -  «A,  (53) 

and  the  final  equations  are 

«>v 

i  =  —       n  c  n  sin  n  (0  —  f)  sin  nr 


and 


where 


=  y  ~  Zjncn  cos  n  (6  —  f) 


cos 


0 


and    T  =  -I. 

^0 


(54) 


(55) 


(3)  Circuit  closed  upon  itself. 

37.  If  a  circuit  of  length  10  is  closed  upon  itself,  then  the  free 
oscillation  of  such  a  circuit  is  characterized  by  the  condition  that 
current  and  voltage  at  I  =  10  are  the  same  as  at  I  =  0,  since  I  =  10 
and  I  =  0  are  the  same  point  of  the  circuit. 

Substituting  this  condition  in  equations  (3)  gives 


=        =       cos        - 


sn 


and 


-  E  =  C,  =  C2  cos  /?Z0  -  j^!  sin  /?Z0; 
herefrom  follows 


(56) 


-  cos 


o,  (i  -  cos 


0)   =    -  yCa  sin  ^0,     > 

=  -  jc,  sin  ^0,    y 


hence, 
or 


-  cos 


=  -        sn 


(57) 


(58) 


NATURAL  PERIOD  OF  TRANSMISSION  LINE 


337 


hence, 


=  2 


(59) 


that  is,  the  circuit  must  be  a  complete  wave   or    a    multiple 
thereof. 

The  free  oscillation  of  a  circuit  which  is  closed  upon  itself  is  a 
full-wave  oscillation ,  containing  a  fundamental  wave  of  frequency 


and  all  the  higher  harmonics  thereof,  the  even  ones  as  well  as  the 
odd  ones, 

/  -  nfr  (61) 

Substituting  in  (3), 


and 

gives 

and 


<7i  =  ci   +  K 


V  2    ~~ 


cos 


= «,  +      i 

=  c'  +  je,"     J 


(c2"  -  yc2 


cos 


sn 


(62) 


Substituting  the  analytic  expression, 

c/  +  jc/'  =  c/  cos  2  TT/^  +  c/'  sin  2  TT/J^,  etc., 
also 


and 
where 


27T 


(63) 


(64) 


that  is,  the  length  of  the  circuit,  I  =  Z0,  is  represented  by  the 
angle  r  =  2  x,  or  a  complete  cycle,  this  gives 


338 


TRANSIENT  PHENOMENA 


/  =  (c/  cos  nd  +  c/'  sin  nd)  cos  nr 

sin  nr 


and 


=  V  n 


(c2"  cos  nO  —  cJ  sin 


or  writing 


:/  cos  nO  +  c2"  sin  nff)  cos  n# 

cos  nd  -  c/  sin  n#)  sin  nr} , 

c/  =  a  cos  n/- 
c/7  =  a  sin  n/- 
c/  =  6  cos  n% 
c"  =  6  sin  n^ 


(65) 


gives 
and 


'  =  a  cos  n  (6  —  -f)  cos  nr  —  b  sin  n  (0  —  %)  sin  nr 

\  -  { b  cos  n  (6  —  %)  cos  nr  -  a  sin  n  (6  -  ;-)  sin  nr  j . 
C 


(66) 


Thus  in  its  most  general  form  the  full-wave  oscillation  gives 
the  equations 


i  =      n  { an  cos  n  (6  —  j-n)  cos  m:  —  bn  sin  n  (0  —%n)  sin  nr 


^ 


where 


~n  cos  nr  - 


0    = 


(0  -  fn)  sin  nr}, 


(67) 


(68) 


and  an,  fn  and  6n,  /n  are  groups  of  four  integration  constants. 

38.  With  a  short  circuit  at  the  end  of  a  transmission  line,  the 
drop  of  potential  along  the  line  varies  fairly  gradually  and 
uniformly,  and  the  instantaneous  rupture  of  a  short  circuit  - 
as  by  a  short-circuiting  arc  blowing  itself  out  explosively  — 


NATURAL  PERIOD  OF  TRANSMISSION  LINE  339 

causes  an  oscillation  in  which  the  lower  frequencies  predominate, 
that  is,  a  low-frequency  high-power  surge.  A  spark  discharge 
from  the  line,  a  sudden  high  voltage  charge  entering  the  line 
locally,  as  directly  by  a  lightning  stroke,  or  indirectly  by  induc- 
tion during  a  lightning  discharge  elsewhere,  gives  a  distribution 
of  potential  which  momentarily  is  very  non-uniform,  changes 
very  abruptly  along  the  line,  and  thus  gives  rise  mainly  to  very 
high  harmonics,  but  as  a  rule  does  not  contain  to  any  appre- 
ciable extent  the  lower  frequencies;  that  is,  it  causes  a  high- 
frequency  oscillation,  more  or  less  local  in  extent,  and  while  of 
high  voltage,  of  rather  limited  power,  and  therefore  less  destruc- 
tive than  a  low-frequency  surge. 

At  the  frequencies  of  the  high-frequency  oscillation  neither 
capacity  nor  inductance  of  the  transmission  line  is  perfectly 
constant:  the  inductance  varies  with  the  frequency,  by  the 
increasing  screening  effect  or  unequal  current  distribution  in 
the  conductor;  the  capacity  increases  by  brush  discharge  over  the 
insulator  surface,  by  the  increase  of  the  effective  conductor 
diameter  due  to  corona  effect,  etc.  The  frequencies  of  the  very 
high  harmonics  are  therefore  not  definite  but  to  some  extent 
variable,  and  since  they  are  close  to  each  other  they  overlap; 
that  is,  at  very  high  frequencies  the  transmission  line  has  no 
definite  frequency  of  oscillation,  but  can  oscillate  with  any 
frequency. 

A  long-distance  transmission  line  has  a  definite  natural  period 
of  oscillation,  of  a  relatively  low  fundamental  frequency  and  its 
overtones,  but  can  also  oscillate  with  any  frequency  whatever, 
provided  that  this  frequency  is  very  high. 

This  is  analogous  to  waves  formed  in  a  body  of  water  of 
regular  shape :  large  standing  waves  have  a  definite  wave  length, 
depending  upon  the  dimensions  of  the  body  of  water,  but  very 
short  waves,  ripples  in  the  water,  can  have  any  wave  length,  and 
do  not  depend  on  the  size  of  the  body  of  water. 

A  further  investigation  of  oscillations  in  conductors  with 
distributed  capacity,  inductance,  and  resistance  requires,  how- 
ever, the  consideration  of  the  resistance,  and  so  leads  to  the 
investigation  of  phenomena  transient  in  space  as  well  as  in  time, 
which  are  discussed  in  Section  IV. 

39.  In  the  equations  discussed  in  the  preceding,  of  the  free 
oscillations  of  a  circuit  containing  uniformly  distributed  resist- 


340  TRANSIENT  PHENOMENA 

ance,  inductance,  capacity ,  and  conductance,  the  energy  losses 
in  the  circuit  have  been  neglected,  and  voltage  and  current 
therefore  appear  alternating  instead  of  oscillating.  That  is, 
these  equations  represent  only  the  initial  or  maximum  values  of 
the  phenomenon,  but  to  represent  it  completely  an  exponential 
function  of  time  enters  as  factor,  which,  as  will  be  seen  in  Section 
IV,  is  of  the  form 

(69) 

where  u  =  ~  (7,  +  f )  mav  be  called  the  "time  constant"  of  the 
2  \C      LI 

circuit. 

While  quarter-wave  oscillations  occasionally  occur,  and  are  of 
serious  importance,  the  occurrence  of  half-wave  oscillations  and 
especially  of  full-wave  oscillations  of  the  character  discussed 
before,  that  is,  of  a  uniform  circuit,  is  less  frequent. 

When  in  a  circuit,  as  a  transmission  line,  a  disturbance  or 
oscillation  occurs  while  this  circuit  is  connected  to  other  cir- 
cuits —  as  the  generating  system  and  the  receiving  apparatus  - 
as  is  usually  the  case,  the  disturbance  generally  penetrates  into 
the  circuits  connected  to  the  circuit  in  which  the  disturbance 
originated,  that  is,  the  entire  system  oscillates,  and  this  oscilla- 
tion usually  is  a  full- wave  oscillation;  that  is,  the  oscillation  of 
a  circuit  closed  upon  itself;  occasionally  a  half- wave  oscillation. 
For  instance,  if  in  a  transmission  system  comprising  generators, 
step-up  transformers,  high-potential  lines,  step-down  trans- 
formers, and  load,  a  short  circuit  occurs  in  the  line,  the  circuit 
comprising  the  load,  the  step-down  transformers,  and  the  lines 
from  the  step-down  transformers  to  the  short  circuit  is  left 
closed  upon  itself  without  power  supply,  and  its  stored  energy  is, 
therefore,  dissipated  as  a  full-wave  oscillation.  Or,  if  in  this 
system  an  excessive  load,  as  the  dropping  out  of  step  of  a  syn- 
chronous converter,  causes  the  circuit  to  open  at  the  generating 
station,  the  dissipation  of  the  stored  energy  —  in  this  case  that 
of  the  excessive  current  in  the  system  —  occurs  as  a  full-wave 
oscillation,  if  the  line  cuts  off  from  the  generating  station  on  the 
low-tension  side  of  the  step-up  transformers,  and  the  oscillating 
circuit  comprises  the  high-tension  coils  of  the  step-up  trans- 
formers, the  transmission  line,  step-down  transformers,  and  load. 
If  the  line  disconnects  from  the  generating  system  on  the  high- 


NATURAL  PERIOD  OF   TRANSMISSION  LINE  341 

potential  side  of  the  step-up  transformers,  the  oscillation  is  a 
half-wave  oscillation,  with  the  two  ends  of  the  oscillating  circuit 
open. 

Such  oscillating  circuits,  however, —  representing  the  most 
frequent  and  most  important  case  of  high-potential  disturbances 
in  transmission  systems, — cannot  be  represented  by  the  preced- 
ing equations  since  they  are  not  circuits  of  uniformly  distributed 
constants  but  complex  circuits  comprising  several  sections  of 
different  constants,  and  therefore  of  different  ratios  of  energy 

consumption   and  energy   storage,  -and  ^-    During  the  free 

Ju  C 

oscillation  of  such  circuits  an  energy  transfer  takes  place  be- 
tween the  different  sections  of  the  circuit,  and  energy  flows  from 
those  sections  in  which  the  energy  consumption  is  small  com- 
pared with  the  energy  storage,  as  transformer  coils  and  highly 
inductive  loads,  to  those  sections  in  which  the  energy  consump- 
tion is  large  compared  with  the  energy  storage,  as  the  more 
non-inductive  parts  of  the  system.  This  introduces  into  the  equa- 
tions exponential  functions  of  the  distance  as  well  as  the  time, 
and  requires  a  study  of  the  phenomenon  as  one  transient  in 
distance  as  well  as  in  time.  The  investigation  of  the  oscillation 
of  a  complex  circuit,  comprising  sections  of  different  constants,  is 
treated  in  Section  IV. 


CHAPTER  IV. 

DISTRIBUTED   CAPACITY   OF    HIGH-POTENTIAL 
TRANSFORMERS. 

40.  In  the  high-potential  coils  of  transformers  designed  for 
very  high  voltages  phenomena  resulting  from  distributed 
capacity  occur. 

In  transformers  for  very  high  voltages  —  100;000  volts  and 
more,  or  even  considerably  less  in  small  transformers  —  the  high- 
potential  coil  contains  a  large  number  of  turns,  a  great  length  of 
conductor,  and  therefore  its  electrostatic  capacity  is  appreciable, 
and  such  a  coil  thus  represents  a  circuit  of  distributed  resistance, 
inductance,  and  capacity  somewhat  similar  to  a  transmission 
line. 

The  same  applies  to  reactive  coils,  etc.,  wound  for  very  high 
voltages,  and  even  in  smaller  reactive  coils  at  very  high  frequency. 

This  capacity  effect  is  more  marked  in  smaller  transformers, 
where  the  size  of  the  iron  core  and  therewith  the  voltage  per 
turn  is  less,  and  therefore  the  number  of  turns  greater  than  in 
very  large  transformers,  and  at  the  same  time  the  exciting  cur- 
rent and  the  full-load  current  are  less;  that  is,  the  charging 
current  of  the  conductor  more  comparable  with  the  load  current 
of  the  transformer  or  reactive  coil. 

However,  even  in  large  transformers  and  at  moderately  high 
voltages,  capacity  effects  occur  in  transformers,  if  the  frequency 
is  sufficiently  high,  as  is  the  case  with  the  currents  produced  in 
overhead  lines  by  lightning  discharges,  or  by  arcing  grounds 
resulting  from  spark  discharges  between  conductor  and  ground, 
or  in  starting  or  disconnecting  the  transformer.  With  such 
frequencies,  of  many  thousand  cycles,  the  internal  capacity  of 
the  transformer  becomes  very  marked  in  its  effect  on  the  dis- 
tribution of  voltage  and  current,  and  may  produce  dangerous 
high-voltage  points  in  the  transformer. 

The  distributed  capacity  of  the  transformer,  however,  is  differ- 
ent from  that  of  a  transmission  line. 

342 


HIGH-POTENTIAL   TRANSFORMERS 


343 


In  a  transmission  line  the  distributed  capacity  is  shunted 
capacity,  that  is,  can  be  represented  diagrammatically  by  con- 
densers shunted  across  the  circuit  from  line  to  line,  or,  what 
amounts  to  the  same  thing,  from  line  to  ground  and  from  ground 
to  return  line,  as  shown  diagrammatically  in  Fig.  88. 


ll.lllUllll-UllJ-  111  II I 
TTTTTTTTTTTTTTTTTTTTTT 


Fig.  88.  Distributed  capacity  of  a  transmission  line. 

0 

The  high-potential  coil  of  the  transformer  also  contains  shunted 
capacity,  or  capacity  from  the  conductor  to  ground,  and  so  each 
coil  element  consumes  a  charging  current  proportional  to  its 
potential  difference  against  ground.  Assuming  the  circuit  as  insu- 
lated, and  the  middle  of  the  transformer  coil  at  ground  potential, 
the  charge  consumed  by  unit  length  of  the  coil  increases  from 
zero  at  the  center  to  a  maximum  at  the  ends.  If  one  terminal 
of  the  circuit  is  grounded,  the  charge  consumed  by  the  coil 
increases  from  zero  at  the  grounded  terminal  to  a  maximum  at 
the  ungrounded  terminal. 

In  addition  thereto,  however,  the  transformer  coil  also  con- 
tains a  capacity  between  successive  turns  and  between  successive 
layers.  Starting  from  one  point  of  the  conductor,  after  a  certain 


C3 

HP 

r 

Hh 

nr 

ii 

II 

IL 

II 
Hi- 

ll 

II 

II 

ii 

ft 

n 

ii 

II 

±* 

-IHHh 

ii 

Tf 

HhHh 

ii 

Hh 

Hi 

-u 

ii 

-ih 

c, 

Hh 

-\[ 

II 
-•tt 

Hh 

Hh 

Hh 

Hh 

Hh 

Hh 

t 

_ci 

111111111. 
ITTTTTTTr 

Till 

•TTTCT 

Fig.  80.    Distributed  capacity  of  a  high -potential  transformer  coil. 


length,  the  length  of  one  turn,  the  conductor  reapproaches  the 
first  point  in  the  next  adjacent  turn.     It  again  approaches  the 


344  TRANSIENT  PHENOMENA 

first  point  at  a  different  and  greater  distance  in  the  next  adjacent 
layer. 

A  transformer  high-potential  coil  can  be  represented  dia- 
grammatically  as  a  conductor,  Fig.  89.  Cl  represents  the  capacity 
against  ground,  C2  represents  the  capacity  between  adjacent 
turns,  and  C3  the  capacity  between  adjacent  layers  of  the  coil. 

The  capacities  C2  and  Ca  are  not  uniformly  distributed  but 
more  or  less  irregularly,  depending  upon  the  number  and  arrange- 
ment of  the  transformer  coils  and  the  number  and  arrangement 
of  turns  in  the  coil.  As  approximation,  however,  the  capacities 
C2  and  (73  can  be  assumed  as  uniformly  distributed  capacity 
between  successive  conductor  elements.  If  I  =  length  of  con- 
ductor, they  may  be  assumed  as  a  capacity  between  I  and  /  +  dl, 
or  as  a  capacity  across  the  conductor  element  dl. 

This  approximation  is  permissible  in  investigating  the  general 
effect  of  the  distributed  capacity,  but  omits  the  effect  of  the 
irregular  distribution  of  C2  and  C3,  which  leads  to  local  oscilla- 
tions of  higher  frequencies,  extending  over  sections  of  the  circuit, 
and  of  lesser  power. 

41.  Let  then,  in  the  high-potential  coil  of  a  high- voltage  trans- 
former, e  =  the  e.m.f.  generated  per  unit  length  of  conductor, 
as,  for  instance,  per  turn;  Z  =  r  — '  jx  =  the  impedance  per  unit 
length;  Y  =  g  —  jb  =  the  capacity  admittance  against  ground 
per  unit  length  of  conductor,  and  Y'  =  pY=  the  capacity 
admittance,  per  unit  length  of  conductor,  between  conductor 
elements  distant  from  each  other  by  unit  length,  as  admittance 
between  successive  turns.  Y'  is  assumed  to  represent  the  total 
effective  admittance  representing  the  capacity  between  successive 
turns,  successive  layers,  and  successive  coils,  as  represented  by 
the  condensers  C2  and  C3  in  Fig.  89. 

The  charging  current  of  a  conductor  element  dl,  due  to  the 
admittance  Y',  is  made  up  of  the  charging  currents  against  the 
next  following  and  that  against  the  preceding  conductor  element. 

Let  1Q  =  length  of  conductor;  I  =  distance  along  conductor; 
E  =  potential  at  point  I,  or  conductor  element  dl,  and  I  =  cur- 
rent in  conductor  element  dl]  then 

dE 

dE  =  —  dl  =  the    potential    difference    between    successive 

Q/i 

conductor  elements  or  turns. 


HIGH-POTENTIAL   TRANSFORMERS  345 

Y7  —  dl  =  the  charging  current  between  one  conductor  ele- 
ment and  the  next  conductor  element  or  turn. 

—  Y'  —  —  77    —  dl  =  the  charging  current  between  one  con- 
dl 

ductor  element  and  the  preceding  conductor  element  or  turn, 
hence, 

dl  =  the  charging  current  of  one  conductor  element  due 


to  capacity  between  adjacent  conductors  or  turns. 

If  now  the  distance  I  is  counted  from  the  point  of  the  con- 
ductor, which  is  at  ground  potential,  YEdl  =  the  charging  cur- 
rent of  one  conductor  element  against  ground,  and 


^Idl 


is  the  total  current  consumed  by  a  conductor  element. 

However,  the  e.m.f.  consumed  by  impedance  equals  the  e.m.f. 
consumed  per  conductor  element;  thus 

dE   =  Zldl 


This  gives  the  two  differential  equations  : 

and  e  -       -  =  ZI.  (2) 

Differentiating  (2)  and  substituting  in  (1)  gives 


transposing, 

-  E 


dP  1  '  (3) 

P     ZY 

ffE 

or  —  =  -  a*E,  (4) 


346 

where 


TRANSIENT  PHENOMENA 

J_ 


a"  = 


/y-j 

P~ZY 


(5) 


1 


If  -==  is  small  compared  with  p,  we  have,  approximately, 


«2  »  -  (6) 

and  E  =  A  cos  aZ  -f  B  sin  aZ,  (7) 

and  since,  for  Z  =  0,  E  =  0,  if  the  distance  Z  is  counted  from  the 
point  of  zero  potential,  we  have 

E  =  B  sin  aZ, 

and  the  current  is  given  by  equation  (2)  as 

1   (          dE  ) 


(8) 
(9) 


substituting  (8)  in  (9)  gives 


I  =  —  \  e  —  aB  cos  aZ 


(10) 


42.   If  now  1 1  =  the  current  at  the  transformer   terminals, 
I  =  Z0,  we  have,  from  (10), 


and 


ZI  j  =  e  —  aB  cos  aZ0 


B  - 


a  cos  a 


substituting  in  (8)  and  (10), 


E  =(e  -  ZI,) 


sin  al 
a  cos  aL 


and 


L  \  cos  al 

f  -1 "  Z  C     •   -     '  l  cosaZ, 


(12) 


for  7j  =  0,  or  open  circuit  of  the  transformer,  this  gives 

sin  al 


E  =  e 


and 


a  cos  aZ( 
cos  aZ 


/  =  e  f cos 
Z  \cos 


(13) 


HIGH-POTENTIAL    TRANSFORMERS  347 

The  e.m.f.,  E,  thus  is  a  maximum  at  the  terminals, 


the  current  a  maximum  at  the  zero  point  of  potential,  I  =  0, 
where 


CHAPTER  V. 

DISTRIBUTED   SERIES  CAPACITY. 

43.  The  capacity  of  a  transmission  line,  cable,  or  high-poten- 
tial transformer  coil  is  shunted  capacity,  that  is,  capacity  from 
conductor  to  ground,  or  from  conductor  to  return  conductor,  or 
shunting  across  a  section  of  the  conductor,  as  from  turn  to  turn 
or  layer  to  layer  of  a  transformer  coil. 

In  some  circuits,  in  addition  to  this  shunted  capacity,  dis- 
tributed series  capacity  also  exists,  that  is,  the  circuit  is  broken 
at  frequent  and  regular  intervals  by  gaps  filled  with  a  dielectric 
or  insulator,  as  air,  and  the  two  faces  of  the  conductor  ends  thus 
constitute  a  condenser  in  series  with  the  circuit.  Where  the 
elements  of  the  circuit  are  short  enough  so  as  to  be  represented, 
approximately,  as  conductor  differentials,  the  circuit  constitutes 
a  circuit  with  distributed  series  capacity. 

An  illustration  of  such  a  circuit' is  afforded  by  the  so-called 
" multi-gap  lightning  arrester,"  as  shown  diagrammatically  in 
Fig.  90,  which  consists  of  a  large  number  of  metal  cylinders  p,  q 
. .  . ,  with  small  spark  gaps  between  the  cylinders,  connected 
between  line  L  and  ground  G.  This  arrangement,  Fig.  90,  can 
be  represented  diagrammatically  by  Fig.  91.  Each  cylinder  has 
a  capacity  (70  against  ground,  a  capacity  C  against  the  adja- 
cent cylinder,  a  resistance  r, —  usually  very  small, —  and  an 
inductance  L. 

If  such  a  series  of  n  equal  spark  gaps  is  connected  across  a 

& 
constant  supply  voltage  e0,  each  gap  has  a  voltage  e  =  — .     If, 

Tl 

however,  the  supply  voltage  is  alternating,  the  voltage  does  not 
divide  uniformly  between  the  gaps,  but  the  potential  difference 
is  the  greater,  that  is,  the  potential  gradient  steeper  the  nearer 
the  gap  is  to  the  line  L,  and  this  distribution  of  potential  becomes 
the  more  non-uniform  the  higher  the  frequency;  that  is,  the 
greater  the  charging  current  of  the  capacity  of  the  cylinder 
against  ground.  The  charging  currents  against  ground,  of  all 

848 


DISTRIBUTED  SERIES  CAPACITY 


349 


the  cylinders  from  q  to  the  ground  G,  Figs.  90  and  91,  must  pass 
the  gap  between  the  adjacent  cylinders  p  and  g;  that  is,  the 
charging  current  of  the  condenser  represented  by  two  adjacent 


-00000000000000-1 


Fig.  90.    Multi-gap  lightning  arrester. 

cylinders  p  and  q  is  the  sum  of  all  the  charging  currents  from 
qtoG',  and  as  the  potential  difference  between  the  two  cylinders 
p  and  q  is  proportional  to  the  charging  current  of  the  condenser 


'filMI — i 


T 


Fig.  91.    Equivalent  circuit  of  a  multi-gap  lightning  arrester. 

formed  by  these  two  cylinders,  C,  this  potential  difference 
increases  towards  L,  being,  at  each  point  proportional  to  the 
vector  sum  of  all  the  charging  currents,  against  ground,  of  all 
the  cylinders  between  this  point  and  ground. 

The  higher  the  frequency,  the  more  non-uniform  is  the  poten- 
tial gradient  along  the  circuit  and  the  lower  is  the  total  supply 
voltage  required  to  bring  the  maximum  potential  gradient,  near 
the  line  L,  above  the  disruptive  voltage,  that  is,  to  initiate  the 
discharge.  Thus  such  a  multigap  structure  is  discriminating 
regarding  frequency;  that  is,  the  discharge  voltage  with  increas- 


350  TRANSIENT  PHENOMENA 

ing  frequency,  does  not  remain  constant,  but  decreases  with 
increase  of  frequency,  when  the  frequency  becomes  sufficiently 
high  to  give  appreciable  charging  currents.  Hence  high  fre- 
quency oscillations  discharge  over  such  a  structure  at  lower 
voltage  than  machine  frequencies. 

For  a  further  discussion  of  the  feature  which  makes  such  a 
multigap  structure  useful  for  lightning  protection,  see  A.  I.  E.  E. 
Transactions,  1906,  pp.  431,  448,  1907,  p.  425,  etc. 

44.  Such  circuits  with  distributed  series  capacity  are  of  great 
interest  in  that  it  is  probable  that  lightning  flashes  in  the  clouds 
are  discharges  in  such  circuits.    From  the  distance  traversed  by 
lightning  flashes  in  the  clouds,  their  character,  and  the  disruptive 
strength  of  air,  it  appears  certain  that  no  potential  difference 
can  exist  in  the  clouds  of  such  magnitude  as  to  cause  a  disruptive 
discharge  across  a  mile  or  more  of  space.     It  is  probable  that 
as  the  result  of  condensation  of  moisture,  and  the  lack  of  uni- 
formity of  such  condensation,  due  to  the  gusty  nature  of  air 
currents,  a  non-uniform  distribution  of  potential  is  produced 
between  the  rain  drops  in  the  cloud;  and  when  the  potential 
gradient  somewhere  in  space  exceeds  the  disruptive  value,  an 
oscillatory  discharge  starts  between  the  rain  drops,  and  grad- 
ually, in  a  number  of  successive  discharges,  traverses  the  cloud 
and   equalizes  the    potential    gradient.     A   study  of   circuits 
containing  distributed  series  capacity  thus  leads  to  an  under- 
standing of  the  phenomena  occurring  in  the  thunder  cloud  during 
the  lightning  discharge.* 

Only  a  general  outline  can  be  given  in  the  following. 

45.  In  a  circuit  containing  distributed  resistance,  conductance, 
inductance,  shunt,  and  series  capacity,  as  the  multigap  lightning 
arrester,  Fig.  90,  represented  electrically  as  a  circuit  in  Fig.  91, 
let  r  =  the  effective  resistance  per  unit  length  of  circuit,  or  per 
circuit  element,  that  is,  per  arrester  cylinder;  g  =  the  shunt 
conductance  per  unit  length,  representing  leakage,  brush  dis- 
charge, electrical  radiation,  etc.;  L  =  the  inductance  per  unit 
length  of  circuit;  C  =  the  series  capacity  per  unit  length  of  cir- 
cuit, or  circuit  element,  that  is,  capacity  between  adjacent  arrester 
cylinders,  and  <70  =  the  shunt  capacity  per  unit  length  of  circuit, 
or  circuit  element,  that  is,  capacity  between  arrester  cylinder  and 

*  See  paper,  "Lightning  and  Lightning  Protection,"  N.E.L.A.,  1907. 
Reprinted  and  enlarged  in  "  General  Lectures  on  Electrical  Engineering,"  by 
Author. 


DISTRIBUTED  SERIES  CAPACITY  351 

ground.     If  then  /  =  the  frequency  of    impressed  e.m.f.,  the 
series  impedance  per  unit  length  of  circuit  is 

Z'=r-j(x-xc);  (1) 

the  shunt  admittance  per  unit  length  of  circuit  is 

Y  -  g  -  jb,  (2) 

where 

x  =  2  nfL, 

1 


b   -  2  xfCt; 
or  the  absolute  values  are 


(3) 


z  =  Vr2  +  (x~xcy 
and  (4) 

y  = 


If  the  distance  along  the  circuit  from  line  L  towards  ground 
G  is  denoted  by  Z,  the  potential  difference  between  point  I  and 
ground  by  E,  and  the  current  at  point  I  by  7,  the  differential 
equations  of  the  circuit  are  * 

f-Z7  (5) 

and 

.   l-3^  <6> 

Differentiating  (5)  and  substituting  (6)  therein  gives 

(7) 

tM 

Equation  (7)  is  integrated  by 

where 

a  =  VYZ  =  a  -  //?,  (9) 


«  =  ^\{yz  +  gr  —  b  (x  —  xc)} 
and  J-  (10) 

/?  = 


*  Section  III,  Chapter  II,  paragraph  7. 


352  TRANSIENT  PHENOMENA 

Substituting  (10)  in  (8)  and  eliminating  the  imaginary  expo- 
nents by  the  substitution  of  trigonometric  functions, 

E  =  A,£-al  (cos  pi  +  j  sin  pi)  +  A2e+al  (cos  pi  -  j  sin  pi).  (11) 

46.  However,  if  n  =  the  total  length  of  circuit  from  line  L  to 
ground  G,  or  total  number  of  arrester  cylinders  between  line  and 
ground,  for  I  =  n, 

E  =  0,  (12) 

and  for  I  =  0, 

E  =  e0  =  the  impressed  e.m.f,  (13) 

Substituting  (12)  and  (13)  into  (11)  gives 

0  =  Ajfi— *  (cos  pn  +  j  sin  pri)  +  A2e+an  (cos  pn-  -  j  sin  pri) 

and 

eo  =  «i  ~t~  ^jJ 

hence, 

1  ""  1  -  £-2an  (cos  2  ph  f  j  sin  2  /?n) '  (14) 

A2  =  -  A^"2"*1  (cos  2pn  +  j  sin  2  /?n), 
and  the  potential  difference  against  ground  is 

c'-a'(cos  pl+j  sin  pi)  -  £-*(2*-f)  [cos  /?  (2  n-Z)  +/  sin  p(2n-l)] 
1  -  £~2*n  (cos  2pn  +  j  sin  2  /?n) 

(15) 
From  equation  (5),  substituting  (15)  and  (9),  we  have 


(cos  pl  +  j  sin  pl)+e~a(2n-l)  [cos  p(2n-l)  +j  sin  ft  (2  n-Z)] 


1  _  £-2aw(cos  2pn  +  j  sin 

(16) 

Reduced  to  absolute  terms  this  gives  the  potential  difference 
against  ground  as 


4/j-s  ll  +  e-2-C2»-o._2e   :  twcos2/?(n  -Z) 
6==eoV r-    -^i^ — o.-2<m^  r^-^: »  (17) 


DISTRIBUTED  SERIES  CAPACITY  353 

the  current  as 


"!     +'  "+2e-:-"cos2/?(tt-0 

1  +«-««-  2.  -»-  COB 


and  the  potential  gradient,  or  potential  difference  between  adja- 
cent cylinders,  is 


For  an  infinite  length  of  line,  n  =  <*>  ,  that  is,  for  a  very  large 
number  of  lightning  arrester  cylinders,  where  e~2an  is  negligible, 
as  in  the  case  where  the  discharge  passes  from  the  line  into  the 
arrester  without  reaching  the  ground,  equations  (17),  (18),  (19) 
simplify  to 

•  -V"1,  (20) 


(21) 

1     £ 

and 

e>  =  e0xc  \Jy-t-*l\  (22) 

that  is,  are  simple  exponential  curves. 
Substituting  (4)  and  (3)  in  (21)  and  (22)  gives 


c  2  + 

(23) 


C2j[l  -  (2^/ 

and 

;  =  27r/Ce';  (24) 

or,  approximately,  if  r  and  </  are  negligible,  we  have 


e' 
and 


-  v cir-- 


47.   Assume,  as  example,  a  lightning  arrester  having  the  fol- 
lowing   constants:    L  =  2  X  10~8    henry;    C0  =  10~13    farads; 


354  TRANSIENT  PHENOMENA 

C  =  4  X  10~n  farads;  r  =  1  ohm;  g  =  4  X  10-6  mho;/  -  108  = 
100  million  cycles  per  second;  n  =  300  cylinders,  and  e0  =  30,000 
volts;  then  from  equation  (3),  x  =  12.6  ohms,  xc  =  39.7  ohms,  and 
b  =  62.8  X  10-6  mhos; 

from  equation  (1), 

Z  =  1  +  27.1  /  ohms; 

from  equation  (2), 

Y  =  (4  -  62.8  j)10-6  mho; 
from  equation  (4), 

z  =  27.1  ohms  and  y  =  62.9  X  10~6  mho; 
from  equation  (10), 

a  =  0.0021  and  /?  =  0.0412; 
from  equation  (17), 

e  =  35,500  Vr0™21  +  0.08  e*-0042'  -  0.568  cos  (24.72  -0.0824  Q; 
from  equation  (18), 

i  =  54  V£-0-0042  '  +  0.08  e^0'0042  1  +  0.568  cos  (24.72  -  0.0824  /), 
and  from  equation  (19), 


e'  -  2140  v^0-0042  '  +  0.08  e^-0042'  +0.568  cos  (24.72  -  0.0824  I)  . 

Hence,  at  I  =  0,  e  =  30,000  volts,  i  -  64.6  amperes,  and 
ef  =  2560  volts;  and  at  I  =  300,  e  =  0,  i  =  57.5  amperes,  arid 
^  =  2280  volts. 

With  voltages  per  gap  varying  from  2280  to  2560,  300  gaps 
would,  by  addition,  give  a  total  voltage  of  about  730,000,  while 
the  actual  voltage  is  only  about  one-twenty-fourth  thereof;  that 
is,  the  sum  of  the  voltages  of  many  spark-gaps  in  series  may  be 
many  times  the  resultant  voltage,  and  a  lightning  flash  may  pass 
possibly  for  miles  through  clouds  with  a  total  potential  of  only 
a  few  hundred  million  volts.  In  the  above  example  the  300 
cylinders  include  7.86  complete  wave-lengths  of  the  discharge. 


CHAPTER  VI. 

ALTERNATING   MAGNETIC   FLUX   DISTRIBUTION. 

48.  As  carrier  of  magnetic  flux  iron  is  used,  as  far  as  possible, 
since  it  has  the  highest  permeability  or  magnetic  conductivity. 
If  the  magnetic  flux  is  alternating  or  otherwise  changing  rapidly, 
an  e.m.f.  is  generated  by  the  change  of  magnetic  flux  in  the  iron, 
and  to  avoid  energy  losses  and  demagnetization  by  the  currents 
produced  by  these  e.m.fs.  the  iron  has  to  be  subdivided  in  the 
direction  in  which  the  currents  would  exist,  that  is,  at  right 
angles  to  the  lines  of  magnetic  force.  Hence,  alternating 
magnetic  fields  and  magnetic  structures  desired  to  respond  very 
quickly  to  changes  of  m.m.f.  are  built  of  thin  wires  or  thin  iron 
sheets,  that  is,  are  laminated. 

Since  the  generated  e.m.fs.  are  proportional  to  the  frequency 
of  the  alternating  magnetism,  the  laminations  must  be  finer 
the  higher  the  frequency. 

To  fully  utilize  the  magnetic  permeability  of  the  iron,  it  there- 
fore has  to  be  laminated  so  as  to  give,  at  the  impressed  frequency, 
practically  uniform  magnetic  induction  throughout  its  section, 
that  is,  negligible  secondary  currents.  This,  however,  is  no 
longer  the  case,  even  with  the  thinnest  possible  laminations, 
at  extremely  high  frequencies,  as  oscillating  currents,  lightning 
discharges,  etc.,  and  under  these  conditions  the  magnetic  flux 
distribution  in  the  iron  is  not  uniform,  but  the  magnetic  flux 
density,  (B,  decreases  rapidly,  and  lags  in  phase,  with  increasing 
depth  below  the  surface  of  the  lamination,  so  that  ultimately 
hardly  any  magnetic  flux  exists  in  the  inside  of  the  laminations, 
but  practically  only  a  surface  layer  carries  magnetic  flux.  The 
apparent  permeability  of  the  iron  thus  decreases  at  very  high 
frequency,  and  this  has  led  to  the  opinion  that  at  very  high  fre- 
quencies iron  cannot  follow  a  magnetic  cycle.  There  is,  however, 
no  evidence  of  such  a  " viscous  hysteresis,"  but  it  is  probable 
that  iron  follows  magnetically  even  at  the  highest  frequencies, 
traversing  practically  the  same  hysteresis  cycle  irrespective  of 

355 


356 


TRANSIENT  PHENOMENA 


the  frequency,  if  the  true  m.m.f.,  that  is,  the  resultant  of  the 
impressed  m.m.f.  and  the  m.m.f.  of  the  secondary  currents  in 
the  iron,  is  considered.  Since  with  increasing  frequency,  at 
constant  impressed  m.m.f.,  the  resultant  m.m.f.  decreases,  due 
to  the  increase  of  the  demagnetizing  secondary  currents,  this 
simulates  the  effect  of  a  viscous  hysteresis. 

Frequently  also,  for  mechanical  reasons,  iron  sheets  of  greater 
thickness  than  would  give  uniform  flux  density  have  to  be  used 
in  an  alternating  field. 

Since  rapidly  varying  magnetic  fields  usually  are  alternating, 
and  the  subdivision  of  the  iron  is  usually  by  lamination,  it  will 
be  sufficient  to  consider  as  illustration  of  the  method  the  dis- 
tribution of  alternating  magnetic  flux  in  iron  laminations. 

49.  Let  Fig.  92  represent  the  section  of  a  lamination.  The 
alternating  magnetic  flux  is  assumed  to  pass  in  a  direction 
perpendicular  to  the  plane  of  the  paper. 

Let  n  =  the  magnetic  permeability,  A  =  the 
electric  conductivity,  I  =  the  distance  of  a  layer 
dl  from  the  center  line  of  the  lamination,  and 
2  10  =  the  total  thickness  of  the  lamination.  If 
then  /  =  the  current  density  in  the  layer  dl, 
and  E  =  the  e.m.f.'per  unit  length  generated  in 
the  zone  dl  by  the  alternating  magnetic  flux,  we 
have 


The  magnetic  flux  density  (Bj  at  the  surface 
I  =  10  of    the  lamination    corresponds    to    the 
Fig.  92.     Alter-   jmpresseci  or  external  m.m.f.      The'  density  (B 

natmg  magnetic  ,,  77  ,  ,, 

fluxdistribution   m  the  zone   dl  corresponds   to  the  impressed 
in  solid  iron.        m.m.f.   plus  the  sum  of  all  the  m.m.fs.  in  the 

zones  outside  of  dl,  or  from  I  to  Z0. 
The  current  in  the  zone  dl  is 


(2) 
(3) 

(4) 


and  produces  the  m.m.f. 

3C=  0.4  TrlE/  dZ, 
which  in  turn  would  produce  the  magnetic  flux  density 


ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION        357 

that  is,  the  magnetic  flux  density  (B  at  the  two  sides  of  the  zone 
dl  differs  by  the  magnetic  flux  density  d&  (equation  (4))  pro- 
duced by  the  m.m.f.  in  zone  dl,  and  this  gives  the  differential 
equation  between  (B,  E,  and  I, 


=  0.4  nlpE.  (5) 

The  e.m.f.  generated  at  distance  I  from  the  center  of  the 
lamination  is  due  to  the  magnetic  flux  in  the  space  from  I  to  1Q. 
Thus  the  e.m.fs.  at  the  two  sides  of  the  zone  dl  differ  from  each 
other  by  the  e.m.f.  generated  by  the  magnetic  flux  ($>dl  in  this 
zone. 

Considering  now  (B,  E,  and  I  as  complex  quantities,  the  e.m.f. 
dE,  that  is,  the  difference  between  the  e.m.fs.  at  the  two  sides  of 
the  zone  dl,  is  in  quadrature  ahead  of  ($>dl,  and  thus  denoted  by 

dE  =   -  j  2  TT/CB  10-8  dl,  (6) 

where  /  =  the  frequency  of  alternating  magnetism. 
This  gives  the  second  differential  equation 

dj[-   -j2^/(BlO-8.  (7) 

50.  Differentiating  (5)  in  respect  to  I,  and  substituting  (7) 
therein,  gives 


.0-8(B,  (8) 

or,  writing 

c2  =  /a2  =  0.4  Tr2/^  10-8,  (9) 

a2  =  0.4  rfXn  10-8,  (10) 

we  have 


- 

This  differential  equation  is  integrated  by 

<B  =  Ac-<";  (12) 

this  equation  substituted  in  (11)  gives 

^=-2^;  (13) 


358  TRANSIENT  PHENOMENA 

hence, 

v  =  ±  (1  -  j)  c  (14) 

and 


Since  ®  must  have  the  same  value  for  —  Z  as  for  +  Z,  being 
symmetrical  at  both  sides  of  the  center  line  of  the  lamination, 

A.  j  :=  A.  2  ==  -A, 

hence, 

CB  =  AO-K'-^'  +  e-'1-^};  (15) 

or,  substituting 


gives 

(B  =  A{(s+c'  +  e~cl)  cos  cZ  -  j  0+cZ-  £~cz)  sin  c/}.       (17) 

51.   Denoting  the  flux  density  in  the  center  of  the  lamination, 
for  I  =  0,  by  (B0  from  (17)  we  have 

(B0  =  2  A; 

hence, 

A  =  i  «.  (18) 

and 


fc  =  <B0  j coscZ  -  y sin  d  > .        (19) 

Denoting  the  flux  density  at  the  outside  of  the  lamination, 
for  Z  =  Z0,  that  is,  the  density  produced  by  the  external  m.m.f., 
by  CBI;  substituted  in  (19),  we  have 

— cos  cZ0  -  y  -  -  sin  cZ0  ( ,     (20) 

£t  Zi  ) 

and  substituting  (20)  in  (19), 

<B  =  A  (<+a  +  <-")  cos  d  -  ;  («+"  -  «-")  and  ;        (21) 

1  (£+ci°  -f  £~c*°)  cos  cZ0—  j  (£+ci°—  e   cl°)  sin  cZ0 

The  mean  or  apparent  value  of  the  flux  density,  i.e.,  the  average 
throughout  the  lamination,  is 

i   ri° 

(22) 


ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION        359 

Using  equation  (15)  as  the  more  convenient  for  integration 
gives 


(23) 


and  substituting  herein  (16),  (18)  and  (20),  gives 

0 —  ) cos  ci  —  j sm  ci  > 

)cLl          2  2  *] 

(24) 

so  /"«>  "^    o c    co\r*n<3/'J   V  (f  "^ c  °  -4-  /?    c'o"\  cjiri  /»/ 

U6.  ^t  t  y  UVJo  Ct<Q       j  v*  i  »  /  •SHI  C*! 


The  absolute  values  of  the  flux  densities  are  derived  as  square 
root  of  the  sum  of  the  squares  of  real  and  imaginary  terms  in 
equations  (19),  (20),  (21),  and  (24),  as 


^2Ve+2d  +  e-*'I  +  2cos2d,  (25) 

o 


CB1  =       >e+2^  +  £-2^o  +  2cos2c/0,  (26) 


(B    = 
and 


2  cos  2 


(28) 


52.  Where  the  thickness  of  lamination,  2  Z0,  or  the  frequency/, 
is  so  great  as  to  give  cZ0  a  value  sufficiently  high  to  make  e~cl°, 
or  the  reflected  wave,  negligible  compared  with  the  main  wave 
e+cl°,  the  equations  can  be  simplified  by  dropping  s~cl.  In  this 
case  the  flux  density,  (B,  is  very  small  or  practically  nothing  in 
the  interior,  and  reaches  appreciable  values  only  near  the  surface. 
It  then  is  preferable  to  count  the  distance  from  the  surface  of  the 


360  TRANSIENT  PHENOMENA 

lamination  into  the  interior,  that  is,  substitute  the  independent 
variable 

s  =  10  -  I.  (29) 

Dropping  €~c*  and  s~cl°  in  equation  (21)  gives 
ecl  (cos  cl  —  f  sin  d) 

/r>     /T>      x          »       / 

• 1  ecl°  (cos  clQ  —  j  sin  clQ) 
=  (B^-^o-^jcosc  (Z0-  1)  +  ysinc  (Z0-  Z)}; 

hence, 

(B  =  ®i  *~cs  (cos  cs  +  y  sin  cs);  (30) 

or  the  absolute  value  is 

(B    =(B1e-c',  (31) 

and  at  the  center  of  the  lamination, 

(B0  =  $1  £~cZ°(cos  cZ0  +  y  sin  cZ0), 


J  sm  cy ,  i 


From  equation  (24)  the  mean  value  of  flux  density  follows 
when  dropping  £~cl°  as  negligible,  thus: 

*-  =  oArf'  (33) 

or  the  absolute  value  is 

(34) 


53.  As  seen,  the  preceding  equations  of  the  distribution  of 
alternating  magnetic  flux  in  a  laminated  conductor  are  of  the 
same  form  as  the  equations  of  distribution  of  current  and  voltage 
in  a  transmission  line,  but  more  special  in  form,  that  is,  the 
attenuation  constant  a  and  the  wave  length  constant  /?  have 
the  same  value,  c.  As  result,  the  distribution  of  the  alternating 
magnetic  flux  in  the  lamina  depends  upon  one  constant  only,  clQ. 

The  wave  length  is  given  by 

cZ    =  2  * 


ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION        361 


i 
hence 

and  by  (9) 


10,000 


and  the  attenuation  during  one  wave  length,  or  decrease  of 
intensity  of  magnetism,  per  wave  length,  is 

£-  2-  =  0.0019, 
and  per  half-wave  length  is 

e-  -  =  0.043. 

At  the  depth  -j  below  the  surface,  the  magnetic  flux  lags  90 
degrees  and  has  considerably  decreased;  at  the  depth  -^  it  lags 

Zi 

180  degrees,  that  is,  is  opposite  in  direction  to  the  flux  at  the 
surface  of  the  lamination,  but  is  very  small,  the  intensity  being 
less  than  5  per  cent  of  that  at  the  surface,  and  at  the  depth  lw 
the  flux  is  again  in  phase  with  the  surface  flux,  but  its  intensity 
is  practically  nil,  less  than  0.2  per  cent  of  the  surface  intensity; 
that  is,  the  penetration  of  alternating  flux  into  the  laminated 
iron  is  inappreciable  at  the  depth  of  one  wave  length. 

By  equations  (33)  and  (34),  the  total  magnetic  flux  per  unit 
width  of  lamination  is 


the  absolute  value  is    2  lQ&m  = 

c" 

that  is,  the  same  as  would  be  produced  at  uniform  density  in  a 
thickness  of  lamination 

2 

or  absolute  value, 


362  TRANSIENT  PHENOMENA 

which  means  that  the  resultant  alternating  magnetism  in  the 
lamination  lags  45  degrees,  or  one-eighth  wave  behind  the  im- 
pressed m.m.f.,  and  is  equal  to  a  uniform  magnetic  density 
penetrating  to  a  depth 

L-~  (36) 


lp,  therefore,  can  be  called  the  depth  of  penetration  of  the 
alternating  magnetism  into  the  solid  iron. 

Since  the  only  constant  entering  into  the  equation  is  cl0,  the 
distribution  of  alternating  magnetism  for  all  cases  can  be  repre- 
sented as  function  of  cl0. 

If  clQ  is  small,  and  therefore  the  density  in  the  center  of  the 
lamination  <B0  comparable  with  the  density  (&1  at  the  outside, 
the  equations  (19),  (20),  and  (24)  respectively  (25),  (26),  and  (28) 
are  preferably  used;  if  cl0  is  large,  and  the  flux  density  B0  in  the 
center  of  the  lamination  is  negligible,  the  equations  (30)  to  (34) 
are  preferably  used. 

54.  As  an  example,  let  /*  =  1000  and  A  =  105;  then  a  =  1.98, 
and  for  /  =  60  cycles  per  second,  c  =  a^/J  =  15.3;  hence,  the 
thickness  of  effective  layer  of  penetration  is 

L  =  —  -  =  0.046  cm  =  0.018  inches. 


In  Fig.  93  is  shown,  with  cl  as  abscissas,  the  effective  value  of 
the  magnetic  flux,  which  from  equation  (25)  is 


&  =  —  °  Ve+'2ci  +  e~'2d  +  2cos2cZ, 

Z 

and  also  the  space-phase  angle  between  (B  and  (B0,  which  from 
equation  (19)  is 

£cl   _  £-cl 

tan  TO  =  ^       __cl  tan  d.  (37) 

In  Fig.  94  is  shown,  with  cs  as  abscissas,  the  effective  value 
of  the  magnetic  flux,  which  from  equation  (31)  is 

CB  =  (B-08 


ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION 


363 


and  also  the  space-phase  angle  between  (B  and  <$>v  which  from 
equation  (30)  is 

tan  TI  =  tan  cs.  (38) 

The  thickness  of  the  equivalent  layer  is  marked  in  Fig.  94. 


\ 

3 

Q 

/ 

I 
1 

100 
80 
GO 
40 

Ofl 

\ 

1 

\ 

o 

o 

Q 

I 

\ 

1 

s 

k 

n 

2 

1 

/ 

\ 

1 

7 

\ 

\ 

1 

i 

' 

/ 

\ 

\ 

/ 

/ 

\  1 

\ 

/ 

/ 

s 

^ 

N 

/ 

, 

t 

\ 

^> 

^ 

3 

0 

^ 

/ 

\ 

/ 

S 

f 

/ 

\ 

/ 

\ 

/ 

X 

y 

cl 

.0        1.6         1.2         0.8         0.4                      0.4         0.8        1.2        1.6         2.0 

*.  93.     Alternating  magnetic  flux  distribution  in  solid  iron. 

As  further   illustrations  are    shown  in  Fig.  95  the  absolute 
values  of  magnetic  flux  density  <&  throughout  a  layer  of  14  mils 
thickness,  that  is,  of  Z0  =  0.007  inches  =  0.018  cm.  thickness. 
For  60  cycles,  by  Curve  I,  c  =    15.3    cZ0  =  0.275 

For  1000  cycles,  by  Curve  II,       c  =    62.5    c/0  =  1.125 
For  10,000  cycles,  by  Curve  III,  c  =  198       cZ0  =  3.55 
It  is  seen  that  the  density  in  Curve  I  is  perfectly  uniform, 
while  in  Curve  III  practically  no  flux  penetrates  to  the  center. 

55.  The  effective  penetration  of  the  alternating  magnetism 
into  the  iron,  or  the  thickness  lp  of  surface  layer  which  at  con- 
stant induction  (Bt  would  give  the  same  total  magnetic  flux  as 
exists  in  the  lamination,  is 

L  =  ,.    1  -   >  (39) 


(1  -  j)c 


or  the  absolute  value  is 


v/2' 


364 


TRANSIENT  PHENOMENA 


1.0 

a 

0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 

\ 

180 
100 
140 
120 

100 
( 

80 

GO 
40 
20 

\ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

V 

A 

\ 

/ 

f 

\ 

/ 

r 

\ 

t 

/ 

\ 

/ 

\ 

/ 

E< 

*~Ti 

uiv. 

icku 

cnt 
ess 

s 

/ 

X 

/ 

\ 

/ 

\j 

/ 

' 

\ 

• 

/ 

x, 

\ 

/ 

—  , 

-  — 

/ 

cs 

^~ 

0.4        0.8        1.2        1.6        2.0        2.4        2.8        3.2 

Fig.  94.     Alternating  magnetic  flux  distribution  in  solid  iron. 


cle's 


1.0 


\ 


ycles 


0.6 


0.4 


0.2 


—  I-I-I-l 


1.0        0.8         0.6         0.4         0.2          0          0.2         0.4         0.6        0.8         1.0 

Fig.  95,     Alternating  magnetic  flux  distribution  in  solid  iron. 


ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION        365 


hence,  substituting  for  c  from  equation  (9), 

104  3570 


(40) 


that  is,  the  penetration  of  an  alternating  magnetic  flux  into  a 
solid  conductor  is  inversely  proportional  to  the  square  root  of 
the  electric  conductivity,  the  magnetic  permeability,  and  the 
frequency. 

The  values  of  penetration,   lp,  in  centimeters   for  various 
materials  and  frequencies  are  given  below. 


Frequency. 

25 

60 

1000 

10,000 

106 

Soft  iron,  n=  1000,  X=  105  
Cast  iron,  /A=  200,     X=  10*  

0.0714 
0  504 

0.0460 
0  325 

0.0113 
0  080 

0.0036 
0  0252 

0.00036 
0  0025 

Copper,  A*=l,  X=6Xl05  

0  922 

0.595 

0  144 

0  0461 

0  0046 

Resistance  alloys,  /*=  1,X=  104 

7.14 

4.60 

1.13 

0.357 

0.036 

As  seen,  even  at  frequencies  as  low  as  25  cycles  alternating 
magnetism  does  not  penetrate  far  into  solid  wrought  iron,  but 
penetrates  to  considerable  depth  into  cast  iron.  It  also  is 
interesting  to  note  that  little  difference  exists  in  the  penetration 
into  copper  and  into  cast  iron,  the  high  conductivity  of  the 
former  compensating  for  the  higher  permeability  of  the  latter. 

56.  The  wave  length,  lw  =  —  ,  substituting  for  c,  from  equa- 


tion (9),  is 


31,600 


(41) 


that  is,  the  wave  length  of  the  oscillatory  transmission  of  alter- 
nating magnetism  in  solid  iron  is  inversely  proportional  to  the 
square  root  of  the  electric  conductivity,  the  magnetic  permea- 
bility, and  the  frequency. 

Comparing  this  equation  (41)  of  the  wave  length  lw  with  equa- 
tion (40)  of  the  depth  of  penetration  lp,  it  follows  that  the  depth 
of  penetration  is  about  one-ninth  of  the  wave  length,  or  40  degrees, 
or,  more  accurately,  since 

1 


IP  = 


27T 

and  l    =  — 


366 
we  have 


TRANSIENT  PHENOMENA 


1 


lw      2  V2n      8.9 
or  40.5  degrees. 
The  speed  of  propagation  is 

31,600  \/J 


(42) 


(43) 


that  is,  the  speed  of  propagation  is  inversely  proportional  to  the 
square  root  of  the  electric  conductivity  and  of  the  magnetic  per- 
meability, but  directly  proportional  to  the  square  root  of  the 
frequency.  This  gives  a  curious  instance  of  a  speed  which 
increases  with  the  frequency.  Numerical  values  are  given  below. 


Frequency. 

25  Cycles. 

10,000 
Cycles. 

Soft  iron, 

C*—  1000   X—  105 

S—  15  8cm 

316  cm 

Cast  iron, 
Copper, 

M-=   200,  A=104  
(tx=       i    A=6  X  105  

111      cm. 

204      cm 

2230  cm. 
4080  cm 

It  is  seen  that  these  speeds  are  extremely  low  compared  with 
the  usual  speeds  of  electromagnetic  waves. 

57.  Since  instead  of  o^,  corresponding  to  the  impressed  m.m.f. 
and  permeability  //,  the  mean  flux  density  in  the  lamina  is  Bm, 
the  effect  is  the  same  as  if  the  permeability  of  the  material  were 
changed  from  /*  to 

//  -  J£®^,  (44) 

and  //  can  be  called  the  effective  permeability,  which  is  a  function 
of  the  thickness  of  the  lamination  and  of  the  frequency,  that  is,  a 
function  of  cZ0;  //  appears  in  complex  form  thus, 

that  is,  the  permeability  is  reduced  and  also  made  lagging. 

For  high  values  of  cl0,  that  is,  thin  laminations  or  high  fre- 
quencies, from  (33),  we  have 


(45) 


ALTERNATING  MAGNETIC  FLUX  DISTRIBUTION        367 

58.  As  illustration,  for  iron  of  14  mils  thickness,  or  10  =  0.018 
centimeters,  and  the  constants  JJL  =  1000  and  A  =  105,  that  is 
a  =  1.98,  the  absolute  value  of  the  effective  permeability  is 


and 
hence, 


(46) 


that  is,  the  effective  or  apparent  permeability  at  very  high 
frequencies  decreases  inversely  proportional  to  the  square  root 
of  the  frequency.  In  the  above  instance  the  apparent  per- 
meability is : 

At  low  frequency,       JJL  =  1000; 

at  10,000  cycles,          //  =  198; 

at  1,000,000  cycles,     //  =  19.8; 

at  100  million  cycles,  //  =  1.98,  and 

at  392  million  cycles,  //  ==  1, 

or  the  same  as  air,  and  at  still  higher  frequencies  the  presence 
of  iron  reduces  the  magnetic  flux. 

It  is  interesting  to  note  that  with  such  a  coarse  lamination 
as  a  14-mil  sheet,  even  at  the  highest  frequencies  of  millions  of 
cycles,  an  appreciable  apparent  permeability  is  still  left;  that  is, 
the  magnetic  flux  is  increased  by  the  presence  of  iron ;  and  the 
effect  of  iron  in  increasing  the  magnetic  flux  disappears  only  at 
400  million  cycles,  and  beyond  this  frequency  iron  lowers  the 
magnetic  flux.  However,  even  at  these  frequencies,  the  presence 
of  iron  still  exerts  a  great  effect  in  the  rapid  damping  of  the 
oscillations  by  the  lag  of  the  mean  magnetic  flux  by  45  degrees. 

Obviously,  in  large  solid  pieces  of  iron,  the  permeability  // 
falls  below  that  of  air  even  at  far  lower  frequencies. 

Where  the  penetration  of  the  magnetic  flux  lp  is  small  com- 
pared with  the  dimensions  of  the  iron,  its  shape  becomes  im- 
material, since  only  the  surface  requires  consideration,  and  so 


368  TRANSIENT  PHENOMENA 

in  this  case  any  solid  structure,  no  matter  what  shape,  can  be 
considered  magnetically  as  its  outer  shell  of  thickness  lp  when 
dealing  with  rapidly  alternating  magnetic  fluxes. 

At  very  high  frequencies,  when  dealing  with  alternating 
magnetic  circuits,  the  outer  surface  and  not  the  section  is,  there- 
fore, the  dominating  feature. 

The  lag  of  the  apparent  permeability  represents  an  energy 
component  of  the  e.m.f.  of  self-induction  due  to  the  magnetic 
flux,  which  increases  with  increasing  frequency,  and  ultimately 
becomes  equal  to  the  reactive  component. 


CHAPTER  VII. 

DISTRIBUTION  OF   ALTERNATING-CURRENT   DENSITY  IN 
CONDUCTOR. 

59.  If  the  frequency  of  an  alternating  or  oscillating  current 
is  high,  or  the  section  of  the  conductor  which  carries  the  current 
is  very  large,  or  its  electric  conductivity  or  its  magnetic  per- 
meability high,  the  current  density  is  not  uniform  throughout 
the  conductor  section,  but  decreases  towards  the  interior  of  the 
conductor,  due  to  the  higher  e.m.f.  of  self-inductance  in  the 
interior  of  the  conductor,  caused  by  the  magnetic  flux  inside  of 
the  conductor.  The  phase  of  the  current  inside  of  the  conductor 
also  differs  from  that  on  the  surface  and  lags  behind  it. 

In  consequence  of  this  unequal  current  distribution  in  a  large 
conductor  traversed  by  ^alternating  currents,  the  effective  resist- 
ance of  the  conductor  may  be  far  higher  than  the  ohmic  resist- 
ance, and  the  conductor  also  contains  internal  inductance. 

In  the  extreme  case,  where  the  current  density  in  the  interior 
of  the  conductor  is  very  much  lower  than  on  the  surface,  or  even 
negligible,  due  to  this  "screening  effect/'  as  it  has  been  called, 
the  current  can  be 'assumed  to  exist  only  in  a  thin  surface  layer 
of  the  conductor,  of  thickness  lp ;  that  is,  in  this  case  the  effective 
resistance  of  the  conductor  for  alternating  currents  equals  the 
ohmic  resistance  of  a  conductor  section  equal  to  the  periphery 
of  the  conductor  times  the  " thickness  of  penetration." 

Where  this  unequal  current  distribution  throughout  the  con- 
ductor section  is  considerable,  the  conductor  section  is  not  fully 
utilized,  but  the  material  in  the  interior  of  the  conductor  is  more 
or  less  wasted.  It  is  of  importance,  therefore,  in  alternating- 
current  circuits,  especially  in  dealing  with  very  large  currents,  or 
with  high  frequency,  or  materials  of  very  high  permeability,  as 
iron,  to  investigate  this  phenomenon. 

An  approximate  determination  of  this  effect  for  the  purpose 
of  deciding  whether  the  unequal  current  distribution  is  so  small 
as  to  be  negligible  in  its  effect  on  the  resistance  of  the  conductor, 

369 


370  TRANSIENT  PHENOMENA 

or  whether  it  is  sufficiently  large  to  require  calculation  and 
methods  of  avoiding  it,  is  given  in  " Alternating-Current  Phe- 
nomena," Chapter  XIV,  paragraph  133. 

An  appreciable  increase  of  the  effective  resistance  over  the 
ohmic  resistance  may  be  expected  in  the  following  cases : 

(1)  In  the  low-tension  distribution  of  heavy  alternating  cur- 
rents by  large  conductors. 

(2)  When  using  iron  as  conductor,  as  for  instance  iron  wires 
in  high  potential  transmissions  for  branch  lines  of  smaller  power, 
or  steel  cables  for  long  spans  in  transmission  lines. 

(3)  In  the  rail  return  of  single-phase  railways. 

(4)  When  carrying  very  high  frequencies,  such  as  lightning 
discharges,  high  frequency  oscillations. 

In  the  last  two  cases,  which  probably  are  of  the  greatest  impor- 
tance, the  unequal  current  distribution  usually  is  such  that 
practically  no  current  exists  at  the  conductor  center,  and  the 
effective  resistance  of  the  track  rail  even  for  25-cycle  alternating 
current  thus  is  several  times  greater  than  the  ohmic  resistance, 
and  conductors  of  low  ohmic  resistance  may  offer  a  very  high 
effective  resistance  to  a  lightning  stroke. 

By  subdividing  the  conductor  into  a  number  of  smaller 
conductors,  separated  by  some  distance  from  each  other,  or  by 
the  use  of  a  hollow  -conductor,  or  a  flat  conductor,  as  a  bar  or 
ribbon,  the  effect  is  reduced,  and  for  high-frequency  discharges, 
as  lightning  arrester  connections,  flat  copper  ribbon  offers  a  very 
much  smaller  effective  resistance  than  a  round  wire.  Strand- 
ing the  conductor,  however,  has  no  direct  effect  on  this  phenom- 
enon, since  it  is  due  to  the  magnetic  action  of  the  current,  and 
the  magnetic  field  in  the  stranded  conductor  is  the  same  as  in 
a  solid  conductor,  other  things  being  equal.  That  is,  while  eddy 
currents  in  the  conductor,  due  to  external  magnetic  fields,  are 
eliminated  by  stranding  the  conductor,  this  is  not  the  case  with 
the  increase  of  the  effective  resistance  by  unequal  current  dis- 
tribution. Stranding  the  conductor,  however,  may  reduce 
unequal  current  distribution  indirectly,  especially  with  iron  as 
conductor  material,  by  reducing  the  effective  or  mean  per- 
meability of  the  conductor,  due  to  the  break  in  the  magnetic 
circuit  between  the  iron  strands,  and  also  by  the  reduction  of 
the  mean  conductivity  of  the  conductor  section.  For  instance, 
if  in  a  stranded  conductor  60  per  cent  of  the  conductor  section 


DISTRIBUTION  OF  ALTERNATING  CURRENT  871 

is  copper,  40  per  cent  space  between  the  strands,  the  mean 
conductivity  is  GO  per  cent  of  that  of  copper.  If  by  the  sub- 
division of  an  iron  conductor  into  strands  the  reluctance  of  the 
magnetic  circuit  is  increased  tenfold,  this  represents  a  reduction 
of  the  mean  permeability  to  one-tenth.  Hence,  if  for  the  con- 
ductor material  proper  n  =  1000,  A  =  105,  and  the  conductor 
section  is  reduced  by  stranding  to  60  per  cent,  the  permeability 
to  one-tenth,  the  mean  values  would  be 

fjL0  =  100    and     ^0  =  0.6  X  105, 

and  the  factor  V7/T,  in  the  equation  of  current  distribution,  is 
reduced  from  VT£  ==  10,000  to  VI^  =  2450,  or  to  24.5  per 
cent  of  its  previous  value.  In  this  case,  however,  with  iron  as 
conductor  material,  an  investigation  must  be  made  on  the  cur- 
rent distribution  in  each  individual  conductor  strand. 

Since  the  simplest  way  of  reducing  the  effect  of  unequal  current 
distribution  is  the  use  of  flat  conductors,  the  most  important  case 
is  the  investigation  of  the  alternating-current  distribution 
throughout  the  section  of  the  flat  conductor.  This  also  gives 
the  solution  for  conductors  of  any  shape  when  the  conductor 
section  is  so  large  that  the  current  penetrates  only  the  surface 
layer,  as  is  the  case  with  a  steel  rail  of  a  single-phase  railway. 
Where  the  alternating  current  penetrates  a  short  distance  only 
into  the  conductor,  compared  with  the  depth  of  penetration  the 
curvature  of  the  conductor  surface  can  be  neglected,  that  is,  the 
conductor  surface  considered  as  a  flat  surface  penetrated  to 
the  same  depth  all  over.  Actually  on  sharp  convex  surfaces  the 
current  penetrates  somewhat  deeper,  somewhat  less  on  sharp 
concave  surfaces,  so  that  the  error  is  more  or  less  compensated. 

60.  In  a  section  of  a  flat  conductor,  as  shown  diagram matically 
in  Fig.  92,  page  356,  let  A  =  the  electric  conductivity  of  conductor 
material;  u  =  the  magnetic  permeability  of  conductor  material; 
I  =  the  distance  counted  from  the  center  line  of  the  conductor, 
and  2  10  =  the  thickness  of  conductor. 

Furthermore,  let  E0  =  the  impressed  e.m.f.  per  unit  length  of 
conductor,  that  is,  the  voltage  consumed  per  unit  length  in  the 
conductor  after  subtracting  the  e.m.f.  consumed  by  the  self- 
inductance  of  the  external  magnetic  field  of  the  conductor;  thus, 
if  El  =  the  total  supply  voltage  per  unit  length  of  conductor 


372  TRANSIENT  PHENOMENA 

and  E2  =  the  external  reactance  voltage,  or  voltage  consumed 
by  the  magnetic  field  outside  of  the  conductor,  between  the  con- 
ductors, we  have 


Let 

7=^4-  ji2  =  current  density  in  conductor  element  dl, 
&  =  ^  +  y&2  =  magnetic  density  in  conductor  element  dl, 
E  =  e.m.f.  consumed  in  the  conductor  element  dl  by  the  self- 
inductance  due  to  the  magnetic  field  inside  of  the  conductor; 
then  the  current  Idl  in  the  conductor  element  represents  the 
m.m.f.  or  field  intensity, 

which  causes  an  increase  of  the  magnetic  density  (B  between  the 
two  sides  of  the  conductor  element  dl  by 


(2) 

The  e.m.f.  consumed  by  self-inductance  is  proportional  to  the 
magnetic  flux  and  to  the  frequency,  and  is  90  time-degrees  ahead 
of  the  magnetic  flux. 

The  increase  of  magnetic  flux  (B  dl,  in  the  conductor  element  dl, 
therefore,  causes  an  increase  in  the  e.m.f.  consumed  by  self- 
inductance  between  the  two  sides  of  the  conductor  element  by 

dE  =  +  2  j7r/te  10-8  dl,  (3) 

where  /  =  the  frequency  of  the  impressed  e.m.f. 

Since  the  impressed  e.m.f.  E0  equals  the  sum  of  the  e.m.f.  con- 
sumed by  self-inductance  E  and  the  e.m.f.  consumed  by  the 

resistance  of  the  conductor  element  -,  we  have 

A 

?.-?+!•  w 

/ 

Differentiating  (4)  gives 

dE=-\dI,  (5) 

A 


DISTRIBUTION  OF  ALTERNATING  CURRENT  373 

and  substituting  (5)  in  (3)  gives 

dl  -    -  2  ]Vr/U&  10~8  dl  (6) 


The  two  differential  equations  (6)  and  (2)  are  in  ®,  7  ,  and  Z, 
which  by  eliminating  &,  give  the  differential  equation  between 
7  and  I:  differentiating  (6)  and  substituting  (2)  therein  gives 


.  (7) 

or  writing 

c2  =  a2/  =  0.4  7T2  10-8  Itf,  (8) 

where 

a2  =  0.4  Tr2  10-8  1?,  (9) 

gives 

^=-•2  ,<•>/.  (10) 

This  differential  equation  (10)  is  integrated  by 

/  -  As-1,  (11) 

and  substituting  (11)  in  (10)  gives 
*-    -2/c2, 

v    =±c(l  r:'i);  (12) 

hence, 

I  =  Af+c(l-W  +  ^2£-c<1-^'.  (13) 

Since  /  gives  the  same  value  for  +1  and  for  —  Z, 

4i  =  42  =  4;  (14) 

hence, 

7  "-^T^-^-f  £~c(l-j}l\.  (15) 

Substituting 

e±^'  =  coscZ  ±  jsincZ  (16) 

gives 

/=  45(e+^  +  e-rf)  coscZ  -  j  (e+cl  -  e~cl)  sincZj,      (17) 

and  for  Z  =  Z0,  or  at  the  conductor  surface, 

Iml  =  A{(e+cl»  +  e-c'»)  cos  cZ0  -  j  (e+cl°  -  £~cl<>)  sincZJ.  (18) 


374  TRANSIENT  PHENOMENA 

At  the  conductor  surface,  however,  no  e.m.f.  of  self-inductance 
due  to  the  internal  field  exists,  and 

/o  -  ^o-  (19) 

Substituting  (19)  in  (18)  gives  the  integration  constant  A,  and 
this  substituted  in  (17)  gives  the  distribution  of  current  density 
throughout  the  conductor  section  as 

+Cl         -oi-+ci-cl 


•  °  (£+c/0  +  s   cl°)  cos  cl0—  j  (s+cl°  -  £~cl°)  sin  cl( 

The  absolute  value  is  given  as  the  square  root  of  the  sum  of 
squares  of  real  and  imaginary  terms, 


The  current  density  in  the  conductor  center,  I  =  0,  is 

2  XE 

j    =  m  _  ^  A^Q 

(g+cio  +  e-e^  cog  ^   _  j  (£+cl0  _  £-cl0j  gin  c 

or  the  absolute  value  is 

2^« 

7      _ 

' 


61.  It  is  seen  that  the  distribution  of  alternating-current 
density  throughout  a  solid  flat  conductor  gives  the  same  equa- 
tion as  the  distribution  of  alternating  magnetic  density  through 
an  iron  rail,  equations  of  the  same  character  as  the  equation  of 
the  long  distance  transmission  line,  but  more  special  in  form. 

The  mean  value  of  current  density  throughout  the  conductor 
section, 

1         f*ln 

(24) 

which  is  derived  in  the  same  manner  as  in  Chapter  V,  §  51,  is 

AE0  { (£+cl(>  -£~c/0)  cos  clQ  --  j  (£+cl°  +  £~cl°)  sin  d0\ 

j  ___      rsr . . 

M»._  g-os)  sinc/0j 
(25) 


DISTRIBUTION  OF  ALTERNATING  CURRENT  375 

and  the  absolute  value  is 

~i     -  -  2  cL     ,     o o«7   *  (26) 


Therefore,  the    increase  of  the  effective   resistance  R  of  the 
conductor  over  the  ohmic  resistance  RQ  is 


cos  cl0  -  j  (g+c*Q-  g-c*Q)  sin  cZ0 


0      (1-j)  cZ0  (e+c'°-  £-^°)  cosc/0  -  j  (£+^+£- 

(28) 
or  the  absolute  value  is 


2  ^  +  £-2^0  _  2  cos  2  cZ0 

62.  If  clQ  is  so  large  that  £~cl°  can  be  neglected  compared  with 
e+cl°,  then  in  the  center  of  the  conductor  /  is  negligible,  and  for 
values  of  I  near  to  Z0,  or  near  the  surface  of  the  conductor,  from 
equation  (20)  we  have 

e  +cl  (cos  d  —  j  sin  d) 
'  °£+cl°  (cosd0  -  j'sincZ0) 
=  IEQ  £c(*~/o)  {cos  c  (Z  -  Z0)  -  j  sin  c  (Z  -  Z0)}. 

Substituting 

^  =  Z0  -  I,  (30) 

where  s  is  the  depth  below  the  conductor  surface,  we  have 

/  =  AEQe~cs  (cos  cs  +  j  sin  cs),  (31) 

and  the  absolute  value  is 

7-^.e—  ;  (32) 

the  mean  value  of  current  density  is 


376  TRANSIENT  PHENOMENA 

and  the  absolute  value  is 

/\±J    Q  ^ 

"  dQV2* 

hence,  the  resistance  ratio,  since  the  current  density  at  the  sur- 
face, or  density  in  the  absence  of  a  screening  effect,  is  1 0  =  AEQ : 

—     ifl       '}d 

R0         I  m 

=  cZ0  -  jdw  (35) 

and  the  absolute  value  is 

§-  =  <*.  ^2;  .  (so) 

that  is,  the  effective  resistance  R  of  the  conductor,  as  given  by 
equation  (28),  and,  for  very  thick  conductors,  from  equation 
(35),  appears  in  the  form 

R  =  RQ  K  -  jwa),  (37) 

which  for  very  thick  conductors  gives  for  ml  and  m2  the  values 


63.  As  the  result  of  the  unequal  current  distribution  in  the 
conductor,  the  effective  resistance  is  increased  from  the  ohmic 
resistance  R  to  the  value 


R  =  R0mv 
R  =  cl0R0, 

and  in  addition  thereto  an  effective  reactance 

X  =  RQm2, 

or 

X  =  d0R0, 

is  produced  in  the  conductor. 

In  the  extreme  case,  where  the  current  does  not  penetrate 
much  below  the  surface  of  the  conductor,  the  effective  resistance 
and  the  effective  reactance  of  the  conductor  are  equal  and  are 


where  Rn  is  the  ohmic  resistance  of  the  conductor. 


DISTRIBUTION  OF  ALTERNATING  CURRENT  377 

It  follows  herefrom  that  only  -r  of  the  conductor  section  is 

clQ 

effective;  that  is,  the  depth  of  the  effective  layer  is 

i  -!..!• 

*~d-~c' 

or,  in  other  words,  the  effective  resistance  of  a  large  conductor 
carrying  an  alternating  current  is  the  resistance  of  a  surface 
layer  of  the  depth 

(39) 

and  in  addition  thereto  an  effective  reactance  equal  to  the 
effective  resistance  results  from  the  internal  magnetic  field  of 
the  conductor. 

Substituting  (8)  in  (39)  gives 


n  VGA 

or 

5030 

~' 


(40) 


It  follows  from  the  above  equations  that  in  such  a  conductor 
carrying  an  alternating  current  the  thickness  of  the  conducting 
layer,  or  the  depth  of  penetration  of  the  current  into  the  con- 
ductor, is  directly  proportional,  and  the  effective  resistance  and 
effective  internal  inductance  inversely  proportional,  to  the  square 
root  of  the  electric  conductivity,  of  the  magnetic  permeability, 
and  of  the  frequency. 

From  equation  (40)  it  follows  that  with  a  change  of  conduc- 
tivity A  of  the  material  the  apparent  conductance,  and  therewith 
the  apparent  resistance  of  the  conductor,  varies  proportionally 
to  the  square  root  of  the  true  conductivity  or  resistivity. 

Curves  of  distribution  of  current  density  throughout  the  sec- 
tion of  the  conductor  are  identical  with  the  curves  of  distribution 
of  magnetic  flux,  as  shown  by  Figs.  93,  94,  95  of  Chapter  VI. 

64.  It  is  interesting  to  calculate  the  depth  of  penetration 
of  alternating  current,  for  different  frequencies,  in  different 
materials,  to  indicate  what  thickness  of  conductor  may  be 
employed. 


378 


TRANSIENT  PHENOMENA 


Such  values  may  be  given  for  25  cycles  and  60  cycles  as  the 
machine  frequencies,  and  for  10,000  cycles  and  1,000,000  cycles 
as  the  limits  of  frequency,  between  which  most  high  frequency 
oscillations,  lightning  discharges,  etc.,  are  found,  and  also  for 
1,000,000,000  cycles  as  about  the  highest  frequencies  which 
can  be  produced.  The  depth  of  penetration  of  alternating 
current  in  centimeters  is  given  below. 


Material 

M 

A 

Penetration  in  cm.  at 

25 

Cycles. 

60 

Cycles. 

10,000 
Cycles. 

106 

Cycles. 

10» 
Cycles. 

Very  soft  iron.  .  . 
Steel  rail  

2000 
1000 
200 

1 
1 
1 
1 
1 

i.ixio5 

105 
10* 

6.2  X  10s 
3.7  X105 
0.33  X  105 
900 
80 
0.2 

10-* 

0.068 
0.101 
0.71 
1.28 
1.65 
5.53 
33.5 
112.5 
2.25  XlO3 
100.6    XlO3 

0.044 
0.065 
0.46 
0.82 
1.07 
3.57 
21.7 
72.7 
1.45  XlO3 
65  XlO3 

3.4  X  lO"3 
5.0  X  10-3 
0.0355 
0.064 
0.082 
0.276 
1.67 
5.63 
112 
5030 

0.34x10- 

0.5    XlO- 
3.55X10- 
6.4    xlO- 
8.2    xlO- 
27.6    X10- 
0.167 
0.563 
11.2 
503 

0.011  X  10- 
0.016X  10- 
0.113X10- 
0.203x10- 
0.263  X  10- 
0.88    XlO- 
5.3      X10~ 
17.9      X10- 
0.36 
16 

Cast  iron  .  .  . 

Copper  

Aluminum  
German  silver  .  . 
Graphite  
Silicon 

Salt  solu.,  cone. 
Pure  river  water 

It  is  interesting  to  note  from  this  table  that  even  at  low 
machine  frequencies  the  depth  of  penetration  in  iron  is  so  little 
as  to  give  a  considerable  increase  of  effective  resistance,  except 
when  using  thin  iron  sheets,  while  at  lightning  frequencies  the 
depth  of  penetration  into  iron  is  far  less  than  the  thickness 
of  sheets  which  can  be  mechanically  produced.  With  copper 
and  aluminum  at  machine  frequencies  this  screening  effect  be- 
comes noticeable  only  with  larger  conductors,  approaching  one 
inch  in  thickness,  but  with  lightning  frequencies  the  effect  is 
such  as  to  require  the  use  of  copper  ribbons  as  conductor,  and 
the  thickness  of  the  ribbon  is  immaterial;  that  is,  increasing  its 
thickness  beyond  that  required  for  mechanical  strength  does  not 
decrease  the  resistance,  but  merely  wastes  material.  In  general, 
all  metallic  conductors,  at  lightning  frequencies  give  such  small 
penetration  as  to  give  more  or  less  increase  of  effective  resistance, 
and  their  use  for  lightning  protection  therefore  is  less  desirable, 
since  they  offer  a  greater  resistance  for  higher  frequencies,  while 
the  reverse  is  desirable. 

Only  pure  river  water  does  not  show  an  appreciable  increase 
of  resistance  even  at  the  highest  obtainable  frequencies,  and 
electrolytic  conductors,  as  salt  solution,  give  no  screening  effect 
within  the  range  of  lightning  frequencies,  while  cast  silicon  can 


DISTRIBUTION  OF  ALTERNATING  CURRENT  379 

even  at  one  million  cycles  be  used  in  a  thickness  up  to  one-half 
inch  without  increase  of  effective  resistance. 

The  maximum  diameter  of  conductor  which  can  be  used  with 
alternating  currents  without  giving  a  serious  increase  of  the 
effective  resistance  by  unequal  current  distribution  is  given 
below. 

At  25  cycles: 

Steel  wire 0.30     cm.  or  0.12  inch 

Copper 2.6      cm.  or  1  inch 

Aluminum 3.3       cm.  or  1.3  inches 

At  60  cycles : 

Steel  wire 0.20   cm.  or  0.08  inch 

Copper 1.6     cm.  or  0.63  inch 

Aluminum 2.1      cm.  or  0.83  inch 

At  lightning  frequencies,  up  to  one  million  cycles : 

Copper 0.013  cm.  or  0.005  inch 

Aluminum 0.016  cm.  or  0.0065  inch 

German  silver 0.055  cm.  or  0.022  inch 

Cast  silicon 1.1       cm.  or  0.44  inch 

Salt  solution 22         cm.  or  8.7  inches 

River  water .  .  All  sizes. 


APPENDIX 

Transient  Unequal  Current  Distribution. 

65.  The  distribution  of  a  continuous  current  in  a  large  con- 
ductor is  uniform,  as  the  magnetic  field  of  the  current  inside 
of  the  conductor  has  no  effect  on  the  current  distribution,  being 
constant.  In  the  moment  of  starting,  stopping,  or  in  any  way 
changing  a  direct  current  in  a  solid  conductor,  the  correspond- 
ing change  of  its  internal  magnetic  field  produces  an  unequal 
current  distribution,  which,  however,  is  transient. 

As  in  this  case  the  distribution  of  current  is  transient  in  time 
as  well  as  in  space,  the  problem  properly  belongs  in  Section  IV, 


380  TRANSIENT  PHENOMENA 

but  may  be  discussed  here,  due  to  its  close  relation  to  the 
permanent  alternating-current  distribution  in  a  solid  conductor. 
Choosing  the  same  denotation  as  in  the  preceding  paragraphs, 
but  denoting  current  and  e.m.f.  by  small  letters  as  instantaneous 
values,  equations  (1),  (2),  and  (4)  of  paragraph  61  remain  the 
same: 

dW  =  OAxidl,  (1) 

,  (2) 


e0=e+-,  (4) 

where  e0  =  voltage  impressed  upon  the  conductor  (exclusive 
of  its  external  magnetic  field)  per  unit  length,  e  =  voltage  con- 
sumed by  the  change  of  internal  magnetic  field,  i  =  current 
density  in  conductor  element  dl  at  distance  I  from  center  line 
of  flat  conductor,  p  =  the  magnetic  permeability  of  the  con- 
ductor, and  \  =  the  electric  conductivity  of  the  conductor. 
Equation  (3),  however, 

dE  =  2  JTcfB  1(T8  dl, 
changes  to 

de  =  -^W-»dl  (3) 

at 

when  introducing  the  instantaneous  values;  that  is,  the  integral 
or  effective  value  of  the  e.m.f.  E  consumed  by  the  magnetic 
flux  density  ®  is  proportional  and  lags  90  time-degrees  behind 
(B,  while  the  instantaneous  value  i  is  proportional  to  the  rate  of 
change  of  (B,  that  is,  to  its  differential  quotient. 
Differentiating  (3)  with  respect  to  dl  gives 


and  substituting  herein  equation  (2)  gives 

§  =-0.4  „„*  10-.  (6) 

Differentiating  (4)  twice  with  respect  to  dl  gives 

ffe      1  tfi 

0  -*•  (7) 


DISTRIBUTION  OF  ALTERNATING  CURRENT  381 

and  substituting  (7)  into  (6)  gives 

'  T°  (8) 


as  the  differential  equation  of  the  current  density  i  in  the  con- 
ductor. 
Substituting 

c2  =  0.4  n^  1(T8  (9) 


gves 


This  equation  (10)  is  integrated  by 

i  =A  +  Be-a2t~bl,  (11) 

and  substituting  (11)  in  (10)  gives  the  relation 


hence, 

b  =  ±  jca,  (12) 

and  substituting  (12)  in  (11),  and  introducing  the  trigonometric 
expressions  for  the  exponential  functions  with  complex  imagi- 
nary exponents, 

i  =  A  +  e-a*'  (C1  cos  cal  +  C2  sin  cal),  (13) 

where 

Cl=Bl  +  B2  and  C2  -  j  (B,  -  5a). 

Assuming  the  current  distribution  as  symmetrical  with  the 
axis  of  the  conductor,  that  is,  i  the  same  for  +  I  and  for  —  I, 
gives 

C2=0; 

hence, 

i  =  A  +  C£~att  cos  cal  (13) 

as  the  equation  of  the  current  distribution  in  the  conductor. 
It  is,  however,  for  t  =  <*>  ,  or  for  uniform  current  distribution, 


382  TRANSIENT  PHENOMENA 

hence,  substituting  in  (13), 

and 


A  =  e 


(14) 


At  the  surface  of  the  conductor,  or  for  I  =  Z0,  no  induction  by 
the  internal  magnetic  field  exists,  but  the  current  has  from  the 
beginning  the  final  value  corresponding  to  the  impressed  e.m.f. 
e0,  that  is,  for  I  =  10, 


and  substituting  this  value  in  (14)  gives 

eQX  =  eQ\  +  Ce~aH  cos  cat 
hence, 


0 


and 


or 


calf 


cos  caL  =  0 


(2  K   -    1)  7T 


a  = 


(2  K   -    1)  7T 

2cL   "~J 


(15) 
(16) 


where  K  is  any  integer. 

There  exists  thus  an  infinite  series  of  transient  terms,  exponen- 
tial in  the  time,  t,  and  trigonometric  in  the  distance,  I,  one  of 
fundamental  frequency,  and  with  it  all  the  odd  harmonics,  and 
the  equation  of  current  density,  from  (14),  thus  is 


;  (2  *  -  1)  catZ 

(2  /c  -  1)  ^ 


where 


2d, 


(17) 


(18) 


The  values  of  the  integration  constants  CK  are  determined 
by  the  terminal  conditions,  that  is,  by  the  distribution  of  current 


DISTRIBUTION  OF  ALTERNATING  CURRENT  383 

density  at  the  moment  of  start  of  the  transient  phenomenon, 
or  t  =  0. 


For  i  =  0, 

,   ^   ^         (2  K  - 


2l» 


n   . 

(19) 


Assuming  that  the  current  density  i0  was  uniform  throughout 
the  conductor  section  before  the  change  of  the  circuit  con- 
ditions which  led  to  the  transient  phenomena  —  as  would  be 
expected  in  a  direct-current  circuit,  —  from  (19)  we  have 

00  (^    If     1  ^     T/ 

2}*  CK  cos  — — — —    -  =  —  (eQX  —  i'0)  =  constant,         (20) 

and  the  coefficients  CK  of  this  Fourier  series  are  derived  in  the 
usual  manner  of  such  series,  thus: 


C.  =  2  avgf- 

L 


(e0A  —  i0)  cos 
4  . 


-;o),  (21) 

7T 

where  avg  [!£(*)]****  denotes  the  average  value  of  the  function 
F(x)  between  the  limits  x=x^  and  x  =  x2 

and  equation  (17)  then  assumes  the  form 


(22) 


This  then  is  the  final  equation  of  the  distribution  of  the 
current  density  in  the  conductor. 

If  now  Zj  =  width  of  the  conductor,  then  the  total  current  in 
the  conductor,  of  thickness  2  1Q,  is 

idl 


or 

(23) 


384  TRANSIENT  PHENOMENA 

For  the  starting  of  current,  that  is,  if  the  current  is  zero, 
t*0  =  0,  in  the  conductor  before  the  transient  phenomenon,  this 
gives 

(24) 


While  the  true  ohmic  resistance,  r0,  per  unit  length  of  the 
conductor  is 

r0  =  m>  (25) 


the  apparent  or  effective  resistance  per  unit  length  of  the  con- 
ductor during  the  transient  phenomenon  is 


£  V  c         1         fi  -  (2 < - i)»oi*  (26) 


and  in  the  first  moment,  for  t  =  0,  is 
since  the  sum  is 

1      '  7T2 


(2*  -  I)2     .8 

The  effective  resistance  of  the  conductor  thus  decreases  from 
oo  at  the  first  moment,  with  very  great  rapidity  —  due  to  the 
rapid  convergence  of  the  series  —  to  its  normal  value. 

66.  As  an  example  may  be  considered  the  apparent  resist- 
ance of  the  rail  ret  irn  of  a  direct-current  railway  during  the 
passage  of  a  car  over  the  track. 

Assume  the  car  moving  in  the  direction  away  from  the 
station,  and  the  current  returning  through  the  rail,  then  the 
part  of  the  rail  behind  the  car  carries  the  full  current,  that  ahead 
of  the  car  carries  no  current,  and  at  the  moment  where  the  car 
wheel  touches  the  rail  the  transient  phenomenon  starts  in  this 
part  of  the  rail.  The  successive  rail  sections  from  the  wheel 
contact  backwards  thus  represent  all  the  successive  stages  of 
the  transient  phenomenon  from  its  start  at  the  wheel  contact 
to  permanent  conditions  some  distance  back  from  the  car. 


DISTRIBUTION  OF  ALTERNATING  CURRENT  385 

Assume  the  rail  section  as  equivalent  to  a  conductor  of 
8  cm.  width  and  8  cm.  height,  or  Zt  =8,  Z0  =  4,  and  the  car 
speed  as  40  miles  per  hour,  or  1800  cm.  per  second. 

Assume  a  steel  rail  and  let  the  permeability  p.  =  1000  and 
the  electric  conductivity  X  =  105. 


Then  c  -  v/0.4  npl  10~8  =  N/1.2566  =  1.121, 


ax2  -  0.122. 

Since  i0  =  0,  the  current   distribution  in  the  conductor,  by 
(22),  is 

+  \  X"  S7T1  ^°>122(2't~1)2'  cos  0.393  (2*  - 

=  e0A  {1-  1.27  [e-0-122*  cos  0.393Z-Je-uo'cosl.l8i  +  t 

cos  1.96  Z  -  +    ...]}, 
the  ohmic  resistance  per  unit  length  of  rail  is 

r0=  Q         =0.156  X  10"6  ohms  per  cm. 
*  V/ 

and  the  effective  resistance  per  unit  length  of  rail,  by  (26),  is 

0.156  X  1Q~6 


At  a  velocity  of  1800  cm.  per  second,  the  distance  from  the 
wheel  contact  to  any  point  p  of  the  rail,  Z',  is  given  as  function 
of  the  time  t  elapsed  since  the  starting  of  the  transient  phenom- 
enon at  point  p  by  the  passage  of  the  car  wheel  over  it,  by  the 
expression  V  =  1800  t,  and  substituting  this  in  the  equation  of 
the  effective  resistance  r  gives  this  resistance  as  function  of  the 
distance  from  the  car,  after  passage, 

=  _  0.156  X  IP"6  _ 

=   1    -  0.81  [fi-**10"1'  +  l£-612xlO-«*'    +  ^£-1700xlO-r+   ^    ^       ] 

ohms  per  cm. 

As  illustration  is  plotted  in  Fig.  96  the  ratio  of  the  effective 
resistance  of  the  rail  to  the  true  ohmic  resistance,  —,  and  with 


386 


TRANSIENT  PHENOMENA 


the  distance  from  the  car  wheel,  in  meters,  as  abscissas,  from  the 

equation 

r  1 


r0     1-0.81 


,-  0.0068 1' 


-  0.0612  /' 


-  0.34  I' 


As  seen  from  the  curve,  Fig.  96,  the  effective  resistance  of  the 
rail  appreciably  exceeds  the  true  resistance  even  at  a  consider- 
able distance  behind  the  car  wheel.  Integrating  the  excess  of 
effective  resistance  over  the  ohmic  resistance  shows  that 


7.0 


6.0 


5.0 


.4.0 


3.0 


2.0 


100  200  300 

Distance  from  Car,  Meters 


Fig.  96.    Transient  resistance  of  a  direct-current  railway  rail  return. 
Car  speed  18  meters  per  second. 


the  excess  of  the  effective  or  transient  resistance  over  the  ohmic 
resistance  is  equal  to  the  resistance  of  a  length  of  rail  of  about 
300  meters,  under  the  assumption  made  in  this  instance,  and  at 
a  car  speed  of  40  miles  per  hour.  This  excess  of  the  transient 
rail  resistance  is  proportional  to  the  car  speed,  thus  less  at  lower 
speeds. 


CHAPTER  VIII. 

VELOCITY  OF   PROPAGATION    OF  ELECTRIC  FIELD. 

67.  In  the  theoretical  investigation  of  electric  circuits  the 
velocity  of  propagation  of  the  electric  field  through  space  is 
usually  not  considered,  but  the  electric  field  assumed  as  instan- 
taneous throughout  space;  that  is,  the  electromagnetic  com- 
ponent of  the  field  is  considered  as  in  phase  with  the  current,  the 
electrostatic  component  as  in  phase  with  the  voltage.  In  reality, 
however,  the  electric  field  starts  at  the  conductor  and  propa- 
gates from  there  through  space  with  a  finite  though  very  high 
velocity,  the  velocity  of  light;  that  is,  at  any  point  in  space 
the  electric  field  at  any  moment  corresponds  not  to  the  condi- 
tion of  the  electric  energy  flow  at  that  moment  but  to  that  at  a 
moment  earlier  by  the  time  of  propagation  from  the  conductor 
to  the  point  under  consideration,  or,  in  other  words,  the  electric 
field  lags  the  more,  the  greater  the  distance  from  the  conductor. 

Since  the  velocity  of  propagation  is  very  high  —  about  3  X  1010 
centimeters  per  second  —  the  wave  of  an  alternating  or  oscillating 
current  even  of  very  high  frequency  is  of  considerable  length ;  at 
60  cycles  the  wave  length  is  0.5  X  109  centimeters,  and  even  at 
a  million  cycles  the  wave  length  is  30,000  centimeters,  or  about 
1000  feet,  that  is,  very  great  compared  with  the  distance  to 
which  electric  fields  usually  extend. 

The  important  part  of  the  electric  field  of  a  conductor  extends 
to  the  return  conductor,  which  usually  is  only  a  few  feet  distant; 
beyond  this,  the  field  is  the  differential  field  of  conductor  and 
return  conductor.  Hence,  the  intensity  of  the  electric  field  has 
usually  already  become  inappreciable  at  a  distance  very  small 
compared  with  the  wave  length,  so  that  within  the  range  in 
which  an  appreciable  field  exists  this  field  is  practically  in  phase 
with  the  flow  of  energy  in  the  conductor,  that  is,  the  velocity  of 
propagation  has  no  appreciable  effect. 

Thus,  the  finite  velocity  of  propagation  of  the  electric  field 
requires  consideration  only: 

387 


388  TRANSIENT  PHENOMENA 

(a)  At  extremely  high  frequencies,  hundreds  of  millions  of 
cycles  per  second,  as  given  by  Hertzian  resonators. 

(6)  In  high  frequency  discharges  having  no  return  circuit  or 
no  well  defined  return  circuit,  as  lightning  discharges.  In  this 
case  the  effective  resistance  of  radiation  may  be  so  large  com- 
pared with  the  ohmic  resistance,  even  when  considering  the 
unequal  current  distribution  in  the  conductor  (Chapter  VII),  that 
the  effect  of  the  conductor  material  practically  disappears.  In 
the  conductors  forming  the  discharge  path  of  lightning  arresters 
this  phenomenon  therefore  requires  serious  consideration. 

(c)  With  high  frequencies,  in  the  case  where  the  field  at  a 
considerable  distance  from  the  conductor  is  of  importance  as  in 
wireless  telegraphy. 

In  wireless  telegraphy  the  electric  field  of  the  sending  antennae 
propagating  through  space  impinges  upon  the  receiving  antennae 
and  there  is  observed  by  its  electromagnetic  and  electrostatic 
effect. 

68.  The  electric  field  of  an  infinitely  long  conductor  without 
return  conductor  decreases  inversely  proportionally  to  the  dis- 
tance, and  therefore  is  represented  by 

^r 

#-j,  CD 

where  ^  is  the  intensity  of  the  electric  field  at  unit  distance  from 
the  conductor. 

The  electric  field  of  a  finite  conductor  of  length  1Q  decreases 
inversely  proportionally  to  the  distance  I  and  also  proportionally 
to  the  angle  subtended  by  the  conductor  10  from  the  distance  /, 
and  since  this  angle,  for  great  distances,  is  inversely  proportional 
to  the  distance  I,  the  electric  field  of  a  finite  conductor  of  length  /0 
without  return  conductor  is  represented  by 


Since  the  electric  field  of  the  return  conductor  is  opposite  to 
that  of  the  conductor,  it  follows  that  the  electric  field  of  an 
infinitely  long  conductor,  with  the  return  conductor  at  distance 
L,  by  equation  (1)  is 

V  V 

*  =  -  77-        -17  >  (3) 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     389 

where  lr  =  ^  cos  r  is  the  projection  of  the  distance  lt  between 
the  conductors  upon  the  direction  I,  that  is,  I'  is  the  difference 
in  the  distance  of  the  two  conductors  from  the  point  I. 
For  large  distances  I,  equation  (3),  becomes 

V9 
#-£•  (4) 

In  the  same  manner,  from  equation  (2)  it  follows  that  the 
decrease  of  the  electric  field  of  the  conductor  of  finite  length  10, 
with  its  return  conductor  at  the  distance  lv  that  is,  of  a  recti- 
linear circuit  of  the  dimensions  of  1Q  and  /t : 

I**  V* 

•(>-&"' >(>+$. 

hence, 

,   y*  ,-v 

f- -p—  (5) 

69.  Since  infinitely  long  conductors,  (1)  and  (4),  are  of 
theoretical  interest  only,  practically  available  are  the  cases  (2) 
and  (5). 

The  electric  field  of  a  closed  circuit  decreases  with  the  cube 
of  the  distance,  hence  much  more  rapidly  than  that  of  a  con- 
ductor without  a  return  conductor,  which  decreases  only  with 
the  square  of  the  distance.  Hence,  where,  as  in  wireless  teleg- 
raphy, action  at  great  distance  is  required,  only  conductors 
without  return  conductor  can  be  used.  To  establish  consider- 
able currents  in  such  open  conductors  requires  high  frequen- 
cies, so  that  the  current  is  absorbed  by  the  capacity  of  the 
conductor  or  the  capacity  attached  to  its  end.  No  conductor  f 
parallel  to  the  ground  can  be  treated  as  conductor  without  : 
return  conductor,  since  secondary  currents  in  the  ground  and  ! 
also  in  the  higher  strata  of  the  atmosphere  act  as  return  con- 
ductor with  regard  to  the  electric  field.  The  practical  reali- 
zation of  a  conductor  without  return  thus  requires  a  vertical 
position  of  the  conductor,  and  for  this  reason  in  wireless  teleg- 
raphy the  vertical  sending  and  receiving  antennae  are  necessary, 
and  the  transmission  is  far  more  successful  across  the  ocean 
than  across  the  land,  since  in  the  latter  case  every  tree,  moun- 


390  TRANSIENT  PHENOMENA 

tain,  etc.,  acts  inductively  as  return  conductor,  and  thus 
increases  the  rapidity  of  the  decrease  of  the  electric  field. 

In  such  a  case  the  use  of  high  frequency  and  of  conductors 
without  return  conductor,  hence  with  electric  fields  decreasing 
relatively  slowly  with  the  distance,  requires  an  introduction  of 
the  velocity  of  propagation  into  the  circuit  equations. 

As  illustrations  will  be  discussed : 

(A)  The  inductance  of  a  finite  section  of  an  infinitely  long  con- 
ductor without  return  conductor. 

(B)  The  mutual  inductance  between  two  finite  conductors 

without  return  conductors,  at  considerable  distance  from 

each  other. 

((7)    The  capacity  of  a  sphere  in  free  space. 
(D)   The  capacity  of  a  sphere  against  ground,  in  space. 

Cases  A  and  B  deal  with  the  electromagnetic,  C  and  D  with  the 
electrostatic  component  of  the  electric  field. 

A.  Inductance  of  a  length  I  of  an  infinitely  long  conductor  without 
return  conductor. 

70.  The  inductance  of  a  length  I  of  a  straight  conductor  is 
usually  given  by  the  equation 

L  =  2Zlog^XlO-9,  (6) 

lr 

where  V  =  the  distance  of  return  conductor,  lr  =  the  radius  of  the 
conductor,  and  the  total  length  of  the  conductor  is  assumed  as 
infinitely  great  compared  with  I  and  I'.  This  is  approximately 
the  case  with  the  conductors  of  a  long  distance  transmission  line. 

For  infinite  distance  lf  of  the  return  conductor,  that  is,  a 
conductor  without  return  conductor,  equation  (6)  gives  L  =  oo ; 
that  is,  a  finite  length  of  an  infinitely  long  conductor  without 
return  conductor  has  an  infinite  inductance  L  and  inversely, 
zero  capacity  C. 

In  equation  (6)  the  magnetic  field  is  assumed  as  instantaneous, 
that  is,  the  velocity  of  propagation  of  the  magnetic  field  is 
neglected.  With  alternating  currents  traversing  the  conductor 
this  is  permissible  when  the  distance  to  the  return  conductor 
is  a  negligible  fraction  of  the  wave  length;  that  is,  if  Z'  is 

§ 

negligible  compared  with  -,  where  S  =  the  speed  of  light  and 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     391 

/  =  the  frequency  of  alternating  current.  It  obviously  is  not 
permissible  in  a  conductor  having  no  return  conductor. 

If  a  conductor  conveying  an  alternating  current  has  no  return 
conductor,  its  circuit  is  closed  by  electrostatic  capacity,  either 
the  distributed  capacity  of  the  conductor  or  capacity  connected 
to  the  ends  of  the  conductor.  To  produce  in  such  a  case  con- 
siderable currents,  either  the  conductor  must  be  very  long  or 
the  frequency  and  e.m.f.  very  high. 

No  conductor  extending  parallel  to  the  ground,  as  a  telegraph 
or  transmission  wire,  can  be  considered  as  having  no  return  con- 
ductor, since  even  if  the  conductor  is  isolated  from  the  ground 
secondary  currents  produced  in  the  ground  (and  in  the  higher 
regions  of  the  atmosphere)  act  inductively  as  return  currents. 

Hence  the  case  of  the  conductor  without  return  conductor  can 
physically  be  realized  only  by  a  conductor  perpendicular  to  the 
ground,  as  the  sending  and  receiving  antennse  of  a  wireless  tele- 
graph station,  and  even  then  completely  only  on  the  ocean, 
where  there  are  no  other  vertical  conductors  in  the  space,  as 
trees,  mountains,  etc.,  which  may  act  as  inductive  returns. 

Since  a  vertical  conductor  is  limited  in  length,  very  high  fre- 
quencies are  required,  and  therefore  the  wave  is  of  moderate 
length,  that  is,  the  velocity  of  propagation  of  the  magnetic  (and 
electrostatic)  field  must  be  considered  when  investigating  the 
self-induction  and  the  mutual  induction  of  such  a  conductor. 

The  magnetic  field  at  a  distance  I  from  the  conductor  and  at 
time  t  corresponds  to  the  current  in  the  conductor  at  the  time 
t  -  t',  where  if  is  the  time  required  for  the  electric  field  to 

travel  the  distance  I,  that  is,  t'  =  -,  where  $  =  the  speed  of  light; 

o 

or,  the  magnetic  field  at  distance  I  and  time  t  corresponds  to  the 
current  in  the  conductor  at  the  time  t  —  -  . 

71.  Representing  the  time  t  by  angle  6  =  2  nft,  where  /== 
the  frequency  of  the  alternating  current  in  the  conductor,  and 
denoting 

2f  Q     _ 

TCj          A   TL  ,^_. 

S  lw 

where 

a 

lw  =  -  =  the  wave  length  of  electric  field, 


392  TRANSIENT  PHENOMENA 

the  field  at  distance  I  and  time  angle  6  corresponds  to  time  angle 
6  —  al,  that  is,  lags  in  time  behind  the  current  in  the  conductor 
by  the  phase  angle  al 
Let 

i    =  I  cos  6  =  current,  absolute  units.  (8) 

The  magnetic  induction  at  distance  I  then  is 

A  =?-^cos(tf  -al);  (9) 

i 

hence,  the  total  magnetic  flux  surrounding  the  conductor,  from 
distance  I  to  infinity  is 


r27 
—  cos  (0  -  al)  dl 


/ 


cos 

prm  fii 


j 


dl  cannot  be  integrated  in  finite  form,  but  represents 

i 

a  new  function  which  in  its  properties  is  intermediate  between 
the  sine  function 


/ 


cos  al  dl  =  —  sin  al 


a 
and  the  logarithmic  function 


and  thus  may  be  represented  by  a  new  symbol, 
sine  —  logarithm  =  sil. 


/sin 
— 


In  the  same  manner        —  —  dl  is  related  to  -  cos  al  and 

a 

to  log  I. 

Introducing  therefore  for  these  two  new  functions  the  symbols 


=  Jf  5^U 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     393 


gives 

$  =  2  7/0  { cos  0  sil  al  +  sin  0  col  a/ } .  (13) 

The  e.m.f.  consumed  by  this  magnetic  flux,  or  e.m.f.  of  induc- 
tance, then  is 


dt 


dd 


hence, 

e  =  4  nfll0  { cos  0  col  al  —  sin  0  sil  al }  ; 

and  since  the  current  is 

i  =  I  cos  0, 


(14) 


the  e.m.f.  consumed  by  the  magnetic  field  beyond  distance  I,  or 
e.m.f.  of  inductance,  contains  a  component  in  phase  with  the 
current,  or  power  component, 

e,  ==  4  TT///O  col  al  cos  0,  (15) 

and  a  component  in  quadrature  with  the  current,  or  reactive  com- 
ponent, 


e2  =  —  4  nfll0  sil  a/  sin  0, 


(16) 


which  latter  leads  the  current  by  a  quarter  period. 

The  reactive  component  e2  is  a  true  self-induction,  that  is,  rep- 
resents a  surging  of  energy  between  the  conductor  and  its  electric 
field,  but  no  power  consumption.  The  effective  component  elt 
however,  represents  a  power  consumption 


p  =  eti 

=  4  nfPl0  col  al  cos2# 


(17) 


by  the  magnetic  field  of  the  conductor,  due  to  its  finite  velocity; 

that  is,  it  represents  the  power  radiated  into  space  by  the  conductor. 

The  energy  component  et  gives  rise  to  an  effective  resistance, 

r  =  %  =  4  ;r/70  col  al,  (18) 

^ 

and  the  reactive  component  gives  rise  to  a  reactance, 

4K/4  sil  al,  (19) 


394  TRANSIENT  PHENOMENA 

When  considering  the  finite  velocity  of  propagation  of  the 
electric  field,  self-inductance  thus  is  not  wattless,  but  contains 
an  energy  component,  and  so  can  be  represented  by  an  impe- 
dance, 

Z  =  r  -  jx 

=  4  TT/TO  (col  al  -  j  sil  al)  1(T9  ohms.  (20) 

The  inductance  would  be  given  by 
I  j  = — - 

=  2  10  { sil  al  +  j  col  al }  10~9  henrys,  (21) 

and  the  power  radiated  by  the  conductor  is 

p  =  i*r. 
72.   The  functions 

Ji         I 
and 


col  al  =  Jt     ^p  dl 

can  in  general  not  be  expressed  in  finite  form,  and  so  have  to  be 
recorded  in  tables.*  Close  approximations  can,  however,  be 
derived  for  the  two  cases  where  /  is"  very  small  and  where  I  is 
very  large  compared  with  the  wave  length  lw  of  the  electric  field, 
and  these  two  cases  are  of  special  interest,  since  the  former  rep- 
resents the  total  magnetic  field  of  the  conductor,  that  is,  its  self- 
inductance,  and  the  latter  the  magnetic  field  interlinked  with  a 
distant  receiving  conductor,  that  is,  the  mutual  inductance 
between  sending  and  receiving  conductor. 
It  is 

silO  =00, 

(22) 

sil  oo  =  0, 
col  00=  0. 

Tables  of  these  and  related  functions  are  given  in  the  appendix,  page  545. 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     395 

And  it  can  be  shown  that  for  small  values  of  al,  that  is,  such 
values  of  I  as  are  only  a  small  fraction  of  a  wave  length,  the 
approximations  hold: 


=  log         -  0-5772, 


col  al  =  -  , 


(23) 


and  for  large  values  of  I,  that  is,  values  of  I  which  make  al  equal 
to  a  considerable  number  of  wave  lengths,  we  have 

2  sin  al         .,    7      2  sin  al 

-  <  sil  al  <  -         T-I 

TTTi!  7T^ 

2  cos  al          i    7      2  cos  al 

-  <  colal  <  -        — , 

TLUi  XH2 

where  n1  and  n2  are  the  two  successive  quadrants  between  which 
al  lies.     For  instance,  for 

al  =  40, 
since 

40  =  25.5  X  ^ , 

2i 

n,  -  25, 
^  =  26, 

sin  a£  =  sin  1.5  X  \  =  +  0.707, 
J 

cos  al  =  cos  1.5  X  \  =  -  0.707, 
2 

and 

0.01725  <   sil  40^  <      0.01805, 

-  0.01805  <  col  40  <  --  0.01725. 

As  seen,  for  larger  values  of  al  sil  al  has  the  same  sign  as  the 
sine  function,  col  al  the  same  sign  as  the  cosine  function. 

73.  From  equations  (20)  and  (21)  then  follow,  for  I  =  lr,  the 
self-inductive  impedance  and  the  self-inductance  of  the  con- 
ductor, where  lr  =  the  radius  of  sending  conductor,  and  since  lr 


396  '     TRANSIENT  PHENOMENA 

is  very  small  compared  with  the  wave  length  lw,  the  values  (23) 
can  be  used,  and  give 
Self  -inductive  impedance  : 

Z  =  4  7r//0  ^  -  j  (log  4  -  0.5772)  j  1(T9  ohms,       (25) 
(  A  alr 

and  effective  self-inductance: 

L  =  2  I,  \  log-i-  -  0.5772  +  j^  \  1(T9  henrys.  (26) 

(       alr  2  ) 

As  an  example  let  a  current  of  i=  100  amperes  be  impressed 
upon  a  sending  antenna  of  /0  =  100  feet  =  3  X  103  centimeters, 
consisting  of  a  cylindrical  conductor  of  radius  lr  =  0.4  inch  =  1 
centimeter,  at  a  frequency  of  /=  200,000  cycles,  then 

lw  =  1.5  X  105 
=  0.94  mile, 
a  =  4.19  X  10~5; 

hence, 

L=  (57.2  +  9.4  j)  10~6  henrys, 
Z=  (11.8  -  71.8  j)  ohms, 

or,  absolute, 

z  =  72.8  ohms. 

Hence,  the  voltage  required  by  i  =  IjOO  amperes  is 

e  =  7280  volts, 
and  the  power  radiated  into  space  during  the  oscillation  is 

p  =  tfr  =  118  kilowatts. 

74.  Since  the  effective  resistance  of  the  total  electromagnetic 
radiation,  from  the  conductor  surface  to  infinity,  is,  by  (25), 

-9,  (27) 


it  follows  that  the  effective  resistance,  of  electromagnetic  radia- 
tion of  a  conductor  is  proportional  to  the  frequency  and  to  the 
length  of  the  conductor,  but  independent  of  its  size  or  shape,  and 
the  radiated  power  is 

p  =  2  Tr2/^  10-»,  (28) 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     397 

or  proportional  to  the  frequency.  Thus  while  the  radiated  power 
is  moderate  at  commercial  frequencies,  it  becomes  considerable 
at  very  high  frequencies,  and  then  requires  consideration. 

For  instance,  at  i  =  100  amperes  per  100  feet  =  3000  centi- 
meters of  conductor,  the  radiated  power  is 

At  60  cycles  .............  3.5  watts; 

At  10,000  cycles  ..........  5.9  kilowatts; 

At  106  cycles    ............  5900  kilowatts. 

The  imaginary  component  of  self-inductance  L,  that  is,  the 
term  in  L  which  represents  the  power  radiation,  is 

Z0?r  10~9  henrys;  (29) 

hence  independent  of  conductor  size,  shape,  and  material,  of  fre- 
quency, current,  etc. 
The  imaginary  or  reactive  component  of  the  impedance, 


x  =  4  nfllog-  -  0.5772   10~9  ohms, 

\  Cwj*  / 

is  approximately,  neglecting  0.5772  against  log  —  ,  and  substitut- 

ed,. 

ing  equation  (7), 

o 

x  =  4  7r/70log  —  ^—  10~9  ohms 


j-  -log/) 
nir  i 


10~9  ohms.  (30) 


Hence,  with  increasing  frequency  /,  the  reactance  x  increases, 
but  less  than  proportional  to  the  frequency,  due  to  the  appearance 
of  the  term  —  log  /in  equation  (30). 

For  instance,  with  the  constants  Z0  =  100  feet  =  3  X  103, 
lr  =  0.4  inch  =  1,  at  the  speed  of  light,  S  =  3  X  1010,  we  have 

/=     102        104     106       108, 
x   =  0.0667    4.94    319    14,550. 

B.  Mutual  inductance  of  two  conductors  of  finite  length  at  con- 
siderable distance  from  each  other. 


398  TRANSIENT  PHENOMENA 

75.  Let  Zj  and  12  be  the  length  of  the  sending  and  of  the  receiv- 
ing conductor  respectively. 

By  equation  (2),  the  electric  field  of  a  conductor  of  length  lv  at 
a  considerable  distance  I,  is  given  by 


hence,  for  current 

i  =  I  cos  6 

CB  =  IJ  C°S/  -  gQ  (32) 

is  the  electromagnetic  component  of  the  field  at  distance  I. 

The  magnetic  flux  intercepted  by  the  receiving  conductor  of 
length  12J  at  distance  ld  from  the  sending  conductor,  and 
assumed  to  be  inductively  parallel  thereto,  then  is 


p.!./  eoe(fl  -a!) 

J^  Z2 


(34) 


n        cos  a^    77  r°°  sin  al  „  ) 

cos  0    ^  -y-  dl  +  sin  0  J^  -y-  dl  j  .   (33) 

By  partial  integration, 

rcos  a£  .  r°°  .  1       cos  aZ 

—  -  —  dl  =  —   /     cos  al  a-  =  —  -  --  a  col  al, 
I  II 

rsin  al  _  T00  1      sin  a^ 

—  —  dl  =  —  J     sin  al  dj  =  —  -  --  f-  a  sil  al; 

hence, 

7  7  T  (        /i  /cos  a^  i    7  \  n  /sin  ald  _  \  ) 

$  =  Z^27  )  cos  ^  f-j-    -a  col  ay  +  sm  ^  f—  p^  +  a  sil  ald\  i 

(35) 
and  the  mutual  inductance  is 

_  \     ./sin  aL  \  ) 

-  a  col  aU  +  jf  —  —  -  -f  asil  ald\  (  10~9  henrys; 

(36) 


cos 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     399 
the  mutual  impedance, 

*  -  a  col  ald)\ 
*  ' 


X  10~9  ohms,  (37) 

or,  absolute, 

sin  ald  .  )  2     (  cos  ald 

+ 


col  a?>=  1.661X10-' 
=1.665X10- 


Ld 

X  10~9  ohms.  (38) 

76.  As  an  example,  let 

^  =  12  =  100  feet 
-  3  X  103, 

ld  -  100  miles  =  15.9  X  106, 
/  =  200,000  cycles; 

hence, 

a  =  4.19  X  10-5, 

ald  =  666 

=  424.5  xj; 

and 

sin  ald  =  sin  0.5  *-  =  0.707, 

Zi 

71 

cos  ald  =  cos  0.5—  =  0.707, 

then,  by  (24), 

=  1.661  X  10~3 

=  1.665  X  10-3, 


400  TRANSIENT  PHENOMENA 

hence,  approximately, 

sil  ald  =  0.1663  X  10~3, 
col  ald  =  0.1663  X  10~3, 

and 

LM  =  (-  0.227  +  1.026  j)  10~9  henrys, 
Z  =  (0.387  +  0.086  /)  10~3  ohms, 

or,  absolute, 

Lm  =  1.051  X  10-'  henrys, 
z  =  0.3964  X  10~3  ohms. 

Hence,  with  an  oscillating  current  of  100  amperes  in  the  sending 
antenna,  the  oscillating  voltage  generated  in  the  receiving  an- 
tenna, 100  miles  distant,  is 

e  =  iz 
=  0.03964  volts. 

C.   Capacity  of  a  sphere  in  space. 

77.  The  electrostatic  field  of  a  sphere  in  free  space  decreases 
with  the  square  of  the  distance  Z;  hence, 

*  =  y>  (39) 

where  lr  =  the  radius  of  the  sphere  and  e  =  the  voltage  of  the 
sphere. 

Therefore,  if  e0  =  E  cos  0  (40) 

is  the  potential  or  voltage  of  the  sphere,  the  electrostatic  field  at 
distance  I  is 


P 

or,  neglecting  lr  compared  with  Z, 

y_lrEax(0-al)^ 


VELOCITY  OF  PROPAGATION  OF  ELECTRIC  FIELD     401 
and  the  potential,  or  voltage  at  distance  I,  is 
e  =   J   Vdl 

_^£~^4  (42) 

hence,  expanded, 


e  =  1$  jeostf  J/-^  dl  +  sintf  f^<«  };  (43) 

by  equations  (34)  this  gives 

i  (        *  /cos  °l  \          *  /sm  al  i\  1 

=  lrE  \  cosdl—  --  acolan  +  smflf—  _  -fasilaH  |  ;   (44) 


e  = 


hence,  for  I  =  lr,  e  =  E  cos  6. 
The  inductive  capacity  thus  is 


e  .   (  /cos  al  A       .  /sin  a£  A  ) 

K  =  w^Tff  =  M  h—  a  co1  az)  +  H-T-+  a  Sl1  a9  1  ;  (45) 

hence,  at  distance  ^0 

,   (  /cos  aZ  _  \        /sin  aL  _  \  ) 

K  =  lr]  (—  =—  °  -  a  col  a/0  )  +  j  (-7—  °  +  a  sil  aU    ;      (46) 
(  ^     ^o  /          ^     ^o  /•) 

or  the  absolute  value  is 

(47) 


^<+  osil<  J  / 

^0 


78.  For  instance,  let  lr  =  10  feet  =  300  centimeters;  Z0  = 
100  miles  =  15.9  X  106  centimeters,  and  /  =  200,000  cycles  per 
second;  then  (see  example  in  section  B)  we  have 

k  =  10.5  X  10-«; 

hence,  with  e0  =  10,000  volts  impressed  upon  the  sending  sphere, 
the  voltage  induced  statically  in  the  receiving  sphere,  at  100  miles 
distance,  is 

e  =  ke0  =  0.105  volts. 

D.  Sphere  at  a  distance  l^  from  ground. 


402  TRANSIENT  PHENOMENA 

79.  If  lr  is  the  radius  of  a  sphere,  at  a  distance  ^  from  ground 
and  at  a  potential  difference  e  from  ground,  the  ground,  as  zero 
potential  surface,  can  be  replaced  by  the  image  of  the  sphere, 
that  is,  by  a  sphere  of  radius  lr)  elevation  —  Z,  and  potential 
difference  —e. 

The  electrostatic  field  of  such  a  system  of  spheres,  at  dis- 
tance Z  and  elevation  Z2,  then  is  the  difference  of  the  fields  of 
the  two  spheres,  thus  : 

f==          elr  ___  elr 

J    2 


hence,  if  Z  is  large  compared  with  ll  and  Z2, 

4  ILLe 


(48) 


and  the  induced  potential,  at  distance  Z  and  elevation  Z2,  is 

2dl.  (49) 


The  potential  difference  of  the  sending  sphere  at  elevation  Zt  is 

e0  =  E  cos  0.  (50) 

This  voltage  e,  at  considerable  distances  Z,  is  very  small;  that 
is,  the  purpose  of  the  condenser  at  the  top  of  wireless  sending  and 
receiving  antennae  seems  not  so  much  to  send  out  or  receive  elec- 
trostatic fields,  as  to  afford  a  capacity  return  for  the  oscillating 
current  in  the  conductor,  and  thus  produce  a  large  current,  hence 
a  powerful  electromagnetic  field. 


CHAPTER   IX. 

HIGH-FREQUENCY   CONDUCTORS. 

80.  As  the  result  of  the  phenomena  discussed  in  the  preceding 
chapters,  conductors  intended  to  convey  currents  of  very  high 
frequency,  as  lightning  discharges,  high  frequency  oscillations  of 
transmission  lines,  the  currents  used  in  wireless  telegraphy,  etc., 
cannot  be  calculated  by  the  use  of  the  constants  derived  at  low 
frequency,  but  effective  resistance  and  inductance,  and  therewith 
the  power  consumed  by  the  conductor,  and  the  voltage  drop, 
may  be  of  an  entirely  different  magnitude  from  the  values  which 
would  be  found  by  using  the  usual  values  of  resistance  and  induc- 
tance. In  conductors  such  as  are  used  in  the  connections  and 
the  discharge  path  of  lightning  arresters  and  surge  protectors,  the 
unequal  current  distribution  in  the  conductor  (Chapter  VII)  and 
the  power  and  voltage  consumed  by  electric  radiation,  due  to  the 
finite  velocity  of  the  electric  field  (Chapter  VIII),  require  con- 
sideration. 

The  true  ohmic  resistance  in  high  frequency  conductors  is 
usually  entirely  negligible  compared  with  the  effective  resistance 
resulting  from  the  unequal  current  distribution,  and  still  greater 
may  be,  at  very  high  frequency,  the  effective  resistance  repre- 
senting the  power  radiated  into  space  by  the  conductor.  The 
total  effective  resistance,  or  resistance  representing  the  power 
consumed  by  the  current  in  the  conductor,  thus  comprises  the 
true  ohmic  resistance,  the  effective  resistance  of  unequal  current 
distribution,  and  the  effective  resistance  of  radiation. 

The  power  consumed  by  the  effective  resistance  of  unequal 
current  distribution  in  the  conductor  is  converted  into  heat  in 
the  conductor,  and  this  resistance  thus  may  be  called  the 
" thermal  resistance"  of  the  conductor,  to  distinguish  it  from  the 
radiation  resistance.  The  power  consumed  by  the  radiation 
resistance  is  not  converted  into  heat  in  the  conductor,  but  is 
dissipated  in  the  space  surrounding  the  conductor,  or  in  any 
other  conductor  on  which  the  electric  wave  impinges.  That  is, 

403 


404  TRANSIENT  PHENOMENA 

at  very  high  frequency,  the  total  power  consumed  by  the  effective 
resistance  of  the  conductor  does  not  appear  as  heating  of  the 
conductor,  but  a  large  part  of  it  may  be  sent  out  into  space  as 
electric  radiation,  which  accounts  for  the  power  exerted  upon 
bodies  near  the  path  of  a  lightning  stroke,  as  "side  discharge." 

The  inductance  is  reduced  by  the  unequal  current  distribution 
in  the  conductor,  which,  by  deflecting  most  of  the  current  into 
the  outer  layer  of  the  conductor,  reduces  or  practically  eliminates 
the  magnetic  field  inside  of  the  conductor.  The  lag  of  the  mag- 
netic field  in  space,  behind  the  current  in  the  conductor,  due  to 
the  finite  velocity  of  radiation,  also  reduces  the  inductance  to 
less  than  that  from  the  conductor  surface  to  a  distance  of  one- 
half  wave.  An  exact  determination  of  the  inductance  is,  how- 
ever, not  possible;  the  inductance  is  represented  by  the  electro- 
magnetic field  of  the  conductor,  and  this  depends  upon  the 
presence  and  location  of  other  conductors,  etc.,  in  space,  on  the 
length  of  the  conductor,  and  the  distance  from  the  return  con- 
ductor. Since  very  high  frequency  currents,  as  lightning  dis- 
charges, frequently  have  no  return  conductor,  but  the  capacity 
at  the  end  of  the  discharge  path  returns  the  current  as  "  dis- 
placement current,"  the  extent  and  distribution  of  the  magnetic 
field  is  indeterminate.  If,  however,  the  conductor  under  con- 
sideration is  a  small  part  of  the  total  discharge  —  as  the  ground 
connection  of  a  lightning  arrester,  a  small  part  of  the  discharge 
path  from  cloud  to  ground  —  and  the  frequency  very  high,  so 
that  the  wave  length  is  relatively  short,  and  the  space  covered  by 
the  first  half  wave  thus  is  known  to  be  free  of  effective  return 
conductors,  the  magnitude  of  the  inductance  can  be  calculated 
with  fair  approximation  by  assuming  the  conductor  as  a  finite 
section  of  a  conductor  without  return  conductor. 

Here  then,  as  in  many  cases,  for  the  two  extremes  —  low  fre- 
quency, where  unequal  current  distribution  and  radiation  are 
negligible,  and  very  high  frequency,  where  the  current  traverses 
only  the  outer  layer  and  the  total  effect,  contained  within  one 
wave  length,  is  within  a  moderate  distance  of  the  conductor  — 
the  constants  can  be  calculated;  but  for  the  intermediary  case, 
of  moderately  high  frequency,  the  conductor  constants  may  be 
anywhere  between  the  two  limits,  i.e.,  the  low  frequency  values 
and  the  values  corresponding  to  an  infinitely  long  conductor 
without  return  conductor. 


HIGH-FREQUENCY  CONDUCTORS  405 

Since,  however,  the  magnitude  of  the  conductor  constants, 
as  derived  from  the  approximate  equations  of  unequal  current 
distribution  and  of  radiation,  are  usually  very  different  from  the 
low  frequency  values,  their  determination  is  of  interest  even  in 
the  case  of  intermediate  frequency,  as  indicating  an  upper  limit 
of  the  conductor  constants. 

81.  Using  the  following  symbols,  namely, 

Z0  =  the  length  of  conductor, 

A  =  the  sectional  area, 

lCi  =  the  circumference  at  conductor  surface,  that  is,  following 
all  the  indentations  of  the  conductor, 

ZCj  =  the  shortest  circumference  of  the  conductor,  that  is,  cir- 
cumference without  following  its  indentations, 

lr  =  the  radius  of  the  conductor, 

ld  =  the  distance  from  the  return  conductor, 
X  =  the  conductivity  of  conductor  material, 

fi.  =  the  permeability  of  conductor  material, 

/  =  the  frequency, 

S  =  the  speed  of  light  =  3  X  1010   cm.,  and  (1) 

a  =  — —  =  the  wave  length  constant, 

o 

the  true  ohmic  resistance  is 


the  ohmic  reactance,  low  frequency  value  is 

*o  =  2  7r/70  1 2  loge  f  +  ^l  10~9  ohms;  (3) 

or,  reduced  to  common  logarithms  by  dividing  by  log  e, 

x0  =  2  TT/Z  f4.6  log^  +  |)  10~9  ohms.  (4) 

\  l>r        ** 

The  equivalent  depth  of  penetration  of  the  current  into  the  con- 
ductor, from  Chapter  VII,  (40),  is 

104  5030 

(5) 


406  TRANSIENT  PHENOMENA 

hence,  the  effective  resistance  of  unequal  current  distribution,  or 
thermal  resistance  of  the  conductor,  is,  approximately, 

(6) 


and  the  effective  reactance  of  the  internal  flux  is 

10-  ohms.  (7) 

The  effective  resistance  resulting  from  the  finite  velocity  of  the 
electric  field,  or  radiation  resistance,  by  assuming  the  conductor 
as  a  section  of  an  infinitely  long  conductor  without  return  con- 
ductor, from  Chapter  VIII,  (25),  is 

r2  =  2  l^flO-*  «  1.97  IJ1Q-*  ohms,  (8) 


and  the  effective  reactance  of  the  external  field  of  finite  section  of  an 
infinitely  long  round  conductor  without  return  conductor,  from 
Chapter  VIII,  (25),  is 

z2  =  4  7r/Z0  flog.  4-  -  0.5772)  10-9.  (9) 

\        aL  I 


Assuming  now  that  the  external  magnetic  field  of  a  conductor 
of  any  shape  is  equal  to  that  of  a  round  conductor  having  the 
same  minimum  circumference,  as  is  approximately  the  case,  that 
is,  substituting 

42  =  2^r  (10) 

in  equation  (9),  and  also  substituting  (1),  gives 
x2  =  4  Tr/J^log^  -  0.5772)  10-9 

=  1.26  Z0/(loge  ^-  -  0.5772)  10-8;  (11) 

or,  reduced  to  common  logarithms  by  dividing  by  log  e,  and 
substituting  for  S, 

z*  -  5 IJ  (1  -  0.547  log  IJ)  10-»  ohms      1 
=  0.547  IJ  (9.15  -  logZCi/)  10-"  ohms.  J 


HIGH-FREQUENCY  CONDUCTORS 


407 


82.   The  total  impedance  of  the  conductor  for  high  frequencies 
is,  therefore, 


rt  =  kty— 10-*-  L98V^1Q- 

v-  A  v-  A 


.jSVy^ 


1.97  Z0/  10~8, 
1.98^ 


10~4  = 

^-0.5772)10-, 

=  0.547  Z0/ (9.15  -log  U-)10-8; 
while  the  conductor  impedance  for  low  frequencies  is 


(13) 


(2-88  log  lf  +  0.314  /<)  io-8. 


(14) 


Although  the  true  ohmic  resistance  r0  is  independent  of  the 
frequency,  the  thermal  resistance  rl  is  proportional  to  the  square 
root  and  the  radiation  resistance  r2  to  the  first  power  of  the  fre- 
quency. With  increasing  frequency,  the  resistance  rl  is  at  first 
appreciable,  while  r2  is  still  negligible,  and  appears  only  at  still 
higher  frequencies,  and  ultimately,  at  the  highest  frequency, 
becomes  the  dominant  factor.  Since  r2  does  not  contain  the 
conductor  dimensions,  it  follows  that  at  very  high  frequencies 
size,  shape,  and  material  of  the  conductor  are  immaterial  in  their 
effect  on  the  total  effective  resistance  r. 

The  low  frequency  reactance,  XQ,  is  proportional  to  the  fre- 
quency; the  internal  reactance,  xv  is  proportional  only  to  the 
square  root  of  the  frequency;  that  is,  with  increasing  frequency 
the  internal  field  of  the  conductor  has  less  and  less  effect  on  its 
reactance.  The  radiation  reactance,  x2,  increases  proportionally 
with  the  frequency  for  moderate  frequencies,  but  for  higher  fre- 
quencies increases  at  a  lesser  rate  as  soon  as  the  negative  term 


408 


TRANSIENT  PHENOMENA 


in  x2  becomes  appreciable;  it  ultimately  reaches  a  maximum  and 
then  decreases  again,  but  the  latter  at  such  high  frequencies  as 
to  be  of  no  practical  importance.  Besides,  for  extremely  high 
frequencies,  thousands  of  millions  of  cycles,  equation  (12)  does 
not  apply,  as  it  is  only  the  first  term  of  a  series,  and  the  further 
terms  begin  to  become  appreciable. 

83.  As  examples,  the  following  may  be  considered: 

(1)  A  copper  wire  No.  4  B.  and  S.  gauge  of  0.204  inch  =  0.518 
centimeter  diameter. 

(2)  An  iron  wire  of  the  same  size,  d  =  0.204  inch  =  0.518 
centimeter. 

(3)  A  copper  ribbon  of  3  inches  width  and  one-eighth  inch 
thickness,  or  the  dimensions  7.6  by  0.317  centimeters. 

(4)  A  wrought-iron  pipe  of  2  inches  =  5.06  centimeters  exter- 
nal diameter  and  one-eighth  inch  =  0.317  centimeter  thickness 
of  walls,  that  is,  of  nearly  the  same  circumference  as  the  copper 
ribbon  in  (3). 

Assuming  the  following  constants : 

copper,  p  =  1,        X  =  6.2  X  10s; 

wrought  iron,          p  =  2000,  ^  =  1.1  X  105, 
we  have 


A 

lc{  =  1C2  = 

Wire  No.  4  B.  and  S.  Gauge. 

3  In.  by  |  In. 
Ribbon. 

2  In.  Pipe,  J  In. 

Copper. 

Iron. 

Copper. 

Iron. 

0.211 
1.63 

0.211 
1.63 

2.40 
15.85 

4.70 

15.85 

lr  -  2^T 

0.259 

0.259 

2.53 

2.53 

r  = 

7.6X10-° 

43X10-° 

0.67X10-° 

1.  94X1Q-6 

*••_ 

0.00314X10-6/ 

6.28X10-8/ 

0.00314X10-6/ 

6.28X10-6/ 

t°  = 

0.0824X10-6/ 

0.0824X10-6/ 

0.0536X10-6/ 

0.0536X10-3/ 

r,  _  ^  = 

0.155X10-°V/ 

16.5xlO-°v7 

0.0159xlO-6\/7 

1.68X10-°V7 

J-- 

0.0197X10-6/ 

0.0197X10-6/ 

0.0197X10~°/ 

0.0197X10~6/ 

0.00547xlO-°/(8.94-log/) 

0.00547X  10-°  /  (7.  95  -log/) 

HIGH-FREQUENCY  CONDUCTORS  409 

In  x°,  the  distance  from  the  return  conductor  has  been 
chosen  as  ld  =  6  feet  =  182  centimeters.  The  values  of  x*  for 
iron  are  not  realized;  they  are  due  to  the  excessive  field  in  the 
conductor,  caused  by  its  high  permeability,  but  can  be  realized 
only  at  extremely  low  frequency  and  small  currents;  at  larger 
currents,  magnetic  saturation  greatly  reduces  the  reactance,  so 
that  in  iron  conductors  the  internal  reactance  is  a  function  of  the 
current  and  decreases  with  increase  of  current.  In  conductors 
of  the  size  above  discussed,  even  at  25  cycles  the  unequal  current 
distribution  in  the  conductor  is  so  great  as  to  make  equation  (14) 
inapplicable,  and  the  reactance  is  given  by 

z°  =  4  7r//0  loge  lf  X  10~9  +  xv  (15) 

ir 

where  xl  is  the  internal  reactance  of  equation  (7). 

84.  In  Fig.  97  are  plotted  values  of  the  resistances  r0,  rv  r2 
and  of  the  reactances  x*,  x2°,  xv  x2,  for  the  four  conductors  illus- 
trated, for  frequencies  from  one  cycle  to  1000  million  cycles. 
As  it  is  not  possible  to  represent  directly  quantities  varying  over 
such  a  wide  range,  in  Fig.  97  as  abscissas  are  used  the  logarithms 
of  the  frequency,  and  as  ordinates  the  logarithms  of  the  ohmic 
resistance  or  reactance  per  meter  =  100  centimeters  length  of 
conductor;  that  is,  a  geometric  scale  is  used.  This  means  that 
each  scale  section  is  a  change  by  factor  10,  and  since,  as  discussed 
above,  these  quantities  can  be  determined  only  in  their  general 
magnitude,  a  value  one  scale  section  below  another  one,  there- 
fore, is  negligible  compared  with  the  latter  one,  being  only  one- 
tenth  of  it. 

The  following  conclusions  may  be  drawn  from  the  curves 
shown  in  Fig.  97. 

(1)  In  copper  wire  No.  4,  the  true  ohmic  resistance  prepon- 
derates up  to  100  cycles.  At  100  cycles  the  reactance  x°  rises 
beyond  the  resistance,  and  the  true  ohmic  resistance  becomes 
negligible  in  the  impedance  at  1000  cycles.  At  3000  cycles  the 
screening  effect,  or  the  unequal  current  distribution  in  the  con- 
ductor, becomes  appreciable  and  increases  its  heating  at  con- 
stant value  of  current.  The  radiation  resistance  r2  would  equal 
the  ohmic  resistance  r0  at  about  500  cycles  if  at  this  low  fre- 
quency the  case  of  an  infinitely  long  conductor  without  return 


410 


TRANSIENT  PHENOMENA 


-10*- 


•10-8 


S7 


Ml 


Copper  Wjire 


2.  Iron   Wi,re 


No.  4  B.  & 


Eg 


S. 


b.  4  B.  &  S. 


Cyc  ea 


Cyc 


& 


MM 

-10-g 


x 


1— 


0 


-10— 


4.  Ir6n  .Pi>e 
2  in. 


Fig.  97.     High-frequency  conductors. 


HIGH-FREQUENCY  CONDUCTORS  411 

conductor  could  be  realized.  At  500,000  cycles,  the  radiation 
resistance  r2  equals  the  external  reactance  x2,  and  since  the 
internal  resistance  rl  at  high  frequencies  is  equal  to  the  internal 
reactance  x1  at  500,000  cycles,  the  total  effective  resistance  r 
equals  the  total  effective  reactance  x,  that  is,  the  current  lags  45 
degrees;  and  at  still  higher  frequencies  the  lag  of  the  current 
decreases  still  further,  and  the  radiation  resistance  preponder- 
ates. 

(2)  In  iron  wire  No.  4,  the  true  ohmic  resistance  r0  ceases  to  be 
the  main  term  even  at  frequencies  below  10  cycles,  and  the 
screening  effect  or  the  unequal  current  distribution  in  the  con- 
ductor is  marked  at  10  cycles.    The  internal  reactance  x*,  which 
corresponds  to  uniform  current  density,  thus  ceases  to  represent 
the  actual  conditions  at  frequencies  even  below  10  cycles.     Up 
to  about  one  million  cycles,  the  internal  resistance  rt  and  internal 
reactance  xl  preponderate.    At  about  one  million   cycles,  all 
four   quantities  —  internal    resistance   rv    representing    power 
converted  into  heat  in  the  conductor,  radiation  resistance  r2, 
representing  power  radiated  by  the  conductor,  internal  reactance 
xv  representing  the  magnetic  field  in  the  conductor,  the  external 
reactance  xv  representing  the  magnetic  field  outside  of  the 
conductor  —  are  approximately  equal  and  the  current  lags  45 
degrees.    Above  one  million  cycles,  radiation  resistance  r2  and 
external  reactance  x2  preponderate,  and  as  they  are  independent 
of  the  conductor  material  above  one  million  cycles,  the  iron  wire 
thus  becomes  nearly  as  good  —  or  poor  —  a  conductor  as  copper 
wire. 

Similar   relations  exist  between  the   larger  conductors   (3) 
and  (4). 

(3)  Three-inch    by  one-eighth  inch  copper  ribbon.      Above 
10  cycles,  the  reactance  is  greater  than  the  resistance,  and  above 
2000  cycles  unequal  current  distribution  is  marked.     At  50,000 
cycles,  the  radiation  resistance  equals  the  external  reactance, 
and  the  current  lags  45  degrees,  and  at  still  higher  frequencies 
the  radiation  resistance  preponderates,  and  the  current  lags  less 
than  45  degrees. 

(4)  Two-inch  iron  pipe,  one-eighth  inch  walls.    The  internal 
reactance  xf  here  has  no  meaning,  as  it  would  correspond  to  a 
2-inch  iron  rod.    The  ohmic  resistance  r0  ceases  to  be  applicable, 
and  unequal  current  distribution  begins  already  at  one  cycle  per 


412  TRANSIENT  PHENOMENA 

second.  At  about  30  cycles  the  external  reactance  rises  beyond 
the  ohmic  resistance ;  at  5000  cycles  beyond  the  internal  reactance 
and  resistance.  At  30,000  cycles  the  current  lags  45  degrees,  and 
less  than  45  degrees  at  higher  frequencies.  .At  100,000  cycles 
r1  and  xl  are  small  compared  with  r2  and  xv  and  the  conductor 
material  thus  ceases  to  have  an  effect  on  the  voltage  drop;  that 
is,  above  100,000  cycles  a  large  iron  conductor  gives  practically 
the  same  voltage  drop,  at  the  same  current,  as  a  copper  conductor 
of  the  same  circumference;  that  is,  iron  is  nearly  as  good  a  con- 
ductor as  copper,  when  considering  a  finite  section  of  an  infinitely 
long  conductor  without  return  conductor,  that  is,  approximately, 
when  dealing  with  oscillatory  high  frequency  discharges,  as 
lightning. 

It  is  interesting  to  note  the  high  power  component  of  impe- 
dance existing  at  high  frequencies  and  mainly  due  to  the  radia- 
tion resistance,  which  causes  a  rapid  decay  of  the  oscillation,  due 
to  the  high  power  factor.  The  internal  constants  r1  and  x1  are 
equal,  and  in  the  most  important  range  of  high  frequencies, 
from  10,000  to  1,000,000  cycles,  the  external  constants  r2  and 
x2  are  not  very  different  from  each  other  and  their  plotted  curves 
intersect  at  some  certain  frequency.  That  is,  at  high  frequen- 
cies, the  power  radiated  into  space  increases  at  such  a  rapid 
rate  that  the  circuit  never  becomes  highly  reactive.  This, 
however,  applies  only  under  the  consideration  assumed  here; 
under  different  conditions  the  radiation  power  may  be  sufficiently 
limited  to  give  a  large  angle  of  lag  of  the  current  and  therefore  a 
slower  decay  of  the  oscillating  discharge,  that  is,  a  more  sustained 
oscillation.  In  general,  however,  these  results  show  that  even 
at  high  frequencies  and  in  iron  conductors  the  angle  of  lag  may 
be  moderate. 

It  is  interesting  to  note  that  with  increasing  frequency  the 
conductor  material  decreases  in  importance,  and  even  soft  iron 
becomes  as  good  as  copper  in  the  voltage  drop  in  the  conductor, 
and  at  still  much  higher  frequencies  even  the  size  and  shape  of 
the  conductor  become  less  important,  and  ultimately  all  con- 
ductors act  practically  alike. 

85.  From  the  data  of  the  preceding  table  and  Fig.  97  the* 
total  effective  resistance,  reactance,  impedance,  and  the  power 
factor  per  meter  length  of  conductor  for  high  frequency  dis- 
charge are  given  on  p.  413. 


HIGH-FREQUENCY  CONDUCTORS 


413 


Wire  No.  4  B.  and  S.  Gauge. 

Copper. 

Iron. 

Frequency  

104 
0.0212 
0.0286 
0.0356 
0.59 
3.6 

105 
0.202 
0.221 
0.299 
0.67 
30 

10" 
1.986 
1.626 
2.57 
0.77 
257 

104 
0.185 
0.168 
0.250 
0.74 
25 

105 
0.717 
0.736 
1.028 
0.70 
103 

10" 
3.62 
3.26 

4.87 
0.75 
487 

Resistance    r  

Reactance,  x  

Impedance   z 

Power  factor  
Voltage  drop  at  100  amperes  

Copper  Ribbon,  3  In. 
by  J  In. 

Wrought-iron  Pipe, 
2  In.  by  |  In. 

Frequency  

104 
0.0199 
0.0219 
0.0296 

105 
0.198 
0.162 
0.256 
0.77 
26 

106 
1.972 
1.072 
2.24 
0.88 
224 

104 
0.0365 
0.0385 
0.0530 
0.69 
5.3 

105 
0.250 
0.214 
0.326 
0.76 
33 

106 
2.14 
1.24 
2.47 
0.87 
247 

Resistance   Y 

Reactance   x 

Impedance   z                        

Power  factor  
Voltage  drop  at  100  amperes  

0.67 
3.0 

As  seen  herefrom,  within  the  range  of  frequencies  from  104  to 
106  cycles  the  power  factor  varies  from  59  per  cent  to  88  per  cent, 
being  higher  at  the  higher  frequencies,  but  there  is  little  differ- 
ence in  the  power  factor  between  iron  and  copper. 

The  difference  between  the  different  conductors  decreases 
with  increase  of  frequency ;  while  at  104  cycles  the  iron  wire  has 
seven  times  the  voltage  drop  of  the  same  size  of  copper  wire, 
at  106  cycles  it  has  only  90  per  cent  more  voltage  drop. 

With  the  large  conductors  the  difference,  in  the  voltage  drop 
between  iron  and  copper  is  very  small;  the  voltage  drop  in  the 
iron  pipe  at  104  cycles  is  only  80  per  cent  greater  than  in  the  cop- 
per ribbon,  while  at  106  cycles  the  difference  has  decreased  to 
10  per  cent,  that  is,  has  practically  become  negligible,  and  the 
iron  pipe  is  practically  as  good  a  conductor  and  has  also 
practically  the  same  power  factor  as  the  copper  ribbon,  so  that 
a  small  increase  in  the"  size  of  the  iron  pipe,  to  2.5  inches  in  diam- 
eter, would  make  it  superior  to  the  3-inch  copper  ribbon  at 
frequencies  of  a  million  cycles  and  over,  about  the  same  at  105 
cycles,  and  very  little  inferior  at  104  cycles. 

This  is  rather  against  the  usual  expectations,  but  is  due  to  the 
preponderance  of  the  radiation  resistance,  and,  therefore,  does 
not  apply,  or  at  least  not  to  the  same  extent,  where  the  radiation 
resistance  is  smaller. 


SECTION  IV 
TRANSIENTS   IN  TIME  AND   SPACE 


TRANSIENTS    IN   TIME    AND    SPACE 


CHAPTER  I. 

GENERAL  EQUATIONS. 

1.  The  energy  relations  of  an  electric  circuit  can  be  charac- 
terized, as  discussed  in  Section  III,  by  the  four  constants, 
namely : 

r  =  effective  resistance,  representing  the  power  or  rate  of 
energy  consumption  depending  upon  the  current,  tfr;  or  the 
power  component  of  the  e.m.f.  consumed  in  the  circuit,  that  is, 
with  an  alternating  current,  the  voltage,  ir,  in  phase  with  the 
current. 

L  =  effective  inductance,    representing   the   energy   storage 

i2L 

depending  upon  the  current,  -  — ,  as  electromagnetic  component 

& 

of  the  electric  field;  or  the  voltage  generated  due  to  the  change 

of  the   current,  L  — ,  that  is,  with  an  alternating  current,  the 
at 

reactive  voltage  consumed  in  the  circuit  -  jxi,  where  x  =  2  nfL 
and  /  =  frequency. 

g  =  effective  (shunted)  conductance,  representing  the  power 
or  rate  of  energy  consumption  depending  upon  the  voltage,  e*g; 
or  the  power  component  of  the  current  consumed  in  the  circuit, 
that  is,  with  an  alternating  voltage,  the  current,  eg,  in  phase  with 
the  voltage. 

C  =  effective    capacity,     representing    the    energy    storage 

e*C 
depending  upon  the  voltage,  — ,  as  electrostatic  component  of 

the  electric  field;  or  the  current  consumed  by  a  change  of  the 

de 

voltage,  C  — ,  that  is,  with  an  alternating  voltage,  the  (leading) 
dt 

reactive  current  consumed  in  the  circuit  -  jbe,  where  6  =  2 
and  /  =  frequency. 

417 


418  TRANSIENT  PHENOMENA 

In  the  investigation  of  electric  circuits,  these  four  constants, 
r,  L,  g,  C,  usually  are  assumed  as  located  separately  from  each 
other,  or  localized.  Although  this  assumption  can  never  be  per- 
fectly correct,  —  for  instance,  every  resistor  has  some  inductance 
and  every  reactor  has  some  resistance,  —  nevertheless  in  most 
cases  it  is  permissible  and  necessary,  and  only  in  some  classes  of 
phenomena,  and  in  some  kinds  of  circuits,  such  as  high-frequency 
phenomena,  voltage  and  current  distribution  in  long-distance, 
high-potential  circuits,  cables,  telephone  circuits,  etc.,  this 
assumption  is  not  permissible,  but  r,  L,  g,  C  must  be  treated  as 
distributed  throughout  the  circuit. 

In  the  case  of  a  circuit  with  distributed  resistance,  inductance, 
conductance,  and  capacity,  as  r,  L,  g,  C,  are  denoted  the  effec- 
tive resistance,  inductance,  conductance,  and  capacity,  respec- 
tively, per  unit  length  of  circuit.  The  unit  of  length  of  the  circuit- 
may  be  chosen  as  is  convenient,  thus :  the  centimeters  in  the  high- 
frequency  oscillation  over  the  multigap  lightning  arrester  circuit, 
or  a  mile  in  a  long-distance  transmission  circuit  or  high-potential 
cable,  or  the  distance  of  the  velocity  of  light,  300,000  km.,  etc. 

The  permanent  values  of  current  and  e.m.f.  in  such  circuits 
of  distributed  constants  have,  for  alternating-current  circuits, 
been  investigated  in  Section  III,  where  it  was  shown  that  they 
can  be  treated  as  transient  phenomena  in  space,  of  the  complex 
variables,  current  /  and  e.m.f.  E. 

Transient  phenomena  in  circuits  with  distributed  constants, 
and,  therefore,  the  general  investigation  of  such  circuits,  leads  to 
transient  phenomena  of  two  independent  variables,  time  t  and 
space  or  distance  /;  that  is,  these  phenomena  are  transient  in 
time  and  in  space. 

The  difficulty  met  in  studying  such  phenomena  is  that  they 
are  not  alternating  functions  of  time,  and  therefore  can  no  longer 
be  represented  by  the  complex  quantity. 

It  is  possible,  however,  to  derive  from  the  constants  of  the 
circuit,  r,  L,  g,  C,  and  without  any  assumption  whatever  regard- 
ing current,  voltage,  etc.,  general  equations  of  the  electric  cir- 
cuits, and  to  derive  some  results  and  conclusions  from  such 
equations. 

These  general  equations  of  the  electric  circuit  are  based  on  the 
single  assumption  that  the  constants  r,  L,  g,  C  remain  constant 
with  the  time  t  and  distance  I,  that  is,  are  the  same  for  every  unit 


GENERAL  EQUATIONS  419 

length  of  the  circuit  or  of  the  section  of  the  circuit  to  which  the 
equations  apply.  Where  the  circuit  constants  change,  as  where 
another  circuit  joins  the  circuit  in  question,  the  integration  con- 
stants in  the  equations  also  change  correspondingly. 

Special  cases  of  these  general  equations  then  are  all  the  phe- 
nomena of  direct  currents,  alternating  currents,  discharges  of 
reactive  coils,  high-frequency  oscillations,  etc.,  and  the  difference 
between  these  different  circuits  is  due  merely  to  different  values 
of  the  integration  constants. 

2.  In  a  circuit  or  a  section  of  a  circuit  containing  distributed 
resistance,  inductance,  conductance,  and  capacity,  as  a  trans- 
mission line,  cable,  high-potential  coil  of  a  transformer,  telephone 
or  telegraph  circuit,  etc.,  let  r  =  the  effective  resistance  per  unit 
length  of  circuit;  L  =  the  effective  inductance  per  unit  length 
of  circuit;  g  =  the  effective  shunted  conductance  per  unit 
length  of  circuit;  C  =  the  effective  capacity  per  unit  length  of 
circuit;  t  =  the  time,  I  =  the  distance,  from  some  starting 
point;  e  =•-  the  voltage,  'and  i  =  the  current  at  any  point  I  and 
at  any  time  t\  then  e  and  i  are  functions  of  the  time  t  and  the  dis- 
tance I. 

In  an  element  dl  of  the  circuit,  the  voltage  e  changes,  by  de, 
by  the  voltage  consumed  by  the  resistance  of  the  circuit  element, 
ri  dl,  and  by  the  voltage  consumed  by  the  inductance  of  the  cir- 

cuit element,  L  —  dl.    Hence, 

de  _  di 


In  this  circuit  element  dl  the  current  i  changes,  by  di,  by  the 
current  consumed  by  the  conductance  of  the  circuit  element, 
gedl,  and  by  the  current  consumed  by  the  capacity  of  the  circuit 

de 
element,  C  —  dl.    Hence, 


Differentiating  (1)  with  respect  to  t  and  (2)  with  respect  to  I, 
and  substituting  then  (1)  into  (2),  gives 


420  TRANSIENT  PHENOMENA 

and  in  the  same  manner, 

+K7.  (4) 


These  differential  equations,  of  the  second  order,  of  current  i 
and  voltage  e  are  identical;  that  is,  in  an  electric  circuit  current 
and  e.m.f.  are  represented  by  the  same  equations,  which  differ 
by  the  integration  constants  only,  which  are  derived  from  the 
terminal  conditions  of  the  problem. 

Equation  (3)  is  integrated  by  terms  of  the  form 

i  =  Ae~al~bt.  (5) 

Substituting  (5)  in  (3)  gives  the  identity 

a2  =  rg  -  (rC  +  gL)  b  +  LCb2 

=  (bL  -  r)  (bC  -  g).  (6) 

In  the  terms  of  the  form  (5)  the  relation  (6)  thus  must  exist 
between  the  coefficients  of  I  and  t. 
Substituting  (5)  into  (1)  gives 


L«-   -»',  (7) 

tit 

and,  integrated, 


a 

The  integration  constant  of  (8)  would  be  a  function  of  t}  and 
since  it  must  fulfill  equation  (4),  must  also  have  the  form  (5) 

T  O 

for  the  special  value  a  =  0,  hence,  by  (6),  b  =  -  or  b  =  -JJand 

L  C 

therefore  can  be  dropped. 

In  their  most  general  form  the  equations  of  the  electric  circuit 
are 

Ane-fl"|-w},  (9) 

\bnL  ~rAne-^l-^\f  (10) 


an2  -  (bnL  -  r]  (bnC  -  g)  =  0,  (11) 


GENERAL  EQUATIONS  421 

where  An  and  an  and  bn  are  integration  constants,  the  last  two 
being  related  to  each  other  by  the  equation  (11). 

3.  These  pairs  of  integration  constants,  An  and  (an,  6n),  are 
determinated  by  the  terminal  conditions  of  the  problem. 

Some  such  terminal  conditions,  for  instance,  are  : 

Current  i  and  voltage  e  given  as  a  function  of  time  at  one 
point  1Q  of  the  circuit  —  at  the  generating  station  feeding  into 
the  circuit  or  at  the  receiving  end  of  the  transmission  line. 

Current  i  given  at  one  point,  voltage  e  at  another  point  — 
as  voltage  at  the  generator  end,  current  at  the  receiving  end  of 
the  line. 

Voltage  given  at  one  point  and  the  impedance,  that  is,  the 

complex  ratio  -  -  —  ,  at  another  point  —  voltage  at  the  gen- 
amperes 

erator  end,  load  at  the  receiving  end  of  the  circuit. 

Current  and  voltage  given  at  one  time  t0  as  function  of  the 
distance  I  —  distribution  of  voltage  and  current  in  the  circuit 
at  the  starting  moment  of  an  oscillation,  etc. 

Other  frequent  terminal  conditions  are: 

Current  zero  at  all  times  at  one  point  Z0  —  the  open  end  of  the 
circuit. 

Voltage  zero  at  all  times  at  one  point  10  —  the  grounded  or  the 
short-circuited  end  of  the  circuit. 

Current  and  voltage,  at  all  times,  at  one  point  Z0  of  the  circuit, 
equal  to  current  and  voltage  at  one  point  of  another  circuit  — 
connecting  point  of  one  circuit  with  another  one. 

As  illustration,  some  of  these  cases  will  be  discussed  below. 

The  quantities  i  and  e  must  always  be  real;  but  since  an  and 
bn  appear  in  the  exponent  of  the  exponential  function,  an  and 
bn  may  be  complex  quantities,  in  which  case  the  integration 
constants  An  must  be  such  complex  quantities  that  by  com- 
bining the  different  exponential  terms  of  the  same  index  n,  that 
is,  corresponding  to  the  different  pairs  of  a  and  b  derived  from 
the  same  equation  (10),  the  imaginary  terms  in  An  and 

bnL  -  r 

An  cancel. 


an 
In  the  exponential  function 


-al-bt 


422  TRANSIENT  PHENOMENA 

writing 

a  =  h  +  jk    and     b  =  p  +  jq,  (12) 

we  have 

£-al-bt    =  e-hl-pt£-j(kl+qt) 

and  the  latter  term  resolves  into  trigonometric  functions  of  the 
angle 

kl  +  qt. 

kl  +  qt  =  constant  (13) 

therefore  gives  the  relation  between  I  and  t  for  constant  phase 
of  the  oscillation  or  alternation  of  the  current  or  voltage. 

With  change  of  time  t  the  phase  thus  changes  in  position  I 
in  the  circuit,  that  is,  moves  along  the  circuit. 

Differentiating  (13)  with  respect  to  t  gives 

**+•••*> 

or 


that  is,  the  phase  of  the  oscillation  or  alternation  moves  along 
the  circuit  with  the  speed  —  *.,  or,  in  other  words, 


(15) 


is  the  speed  of  propagation  of  the  electric  phenomenon  in  the  cir- 
cuit. 

(If  no  energy  losses  occur,  r  =  0,  g  =  0,  in  a  straight  con- 
ductor in  a  medium  of  unit  magnetic  and  dielectric  constant, 
that  is,  unit  permeability  and  unit  inductive  capacity,  S  is  the 
velocity  of  light.) 

4.  Since  (11)  is  a  quadratic  equation,  several  pairs  or  corre- 
sponding values  of  a  and  b  exist,  which,  in  the  most  general  case, 
are  complex  imaginary.  The  terms  with  conjugate  complex 
imaginary  values  of  a  and  b  then  have  to  be  combined  for  the 
elimination  of  their  imaginary  form,  and  thereby  trigonometric 
functions  appear;  that  is,  several  terms  in  the  equations  (9)  and 


GENERAL  EQUATIONS 


423 


(10),  which  correspond  to  the  same  equation  (11),  and  thus  can 
be  said  to  form  a  group,  can  be  combined  with  each  other. 

Such  a  group  of  terms,  of  the  same  index  n,  is  defined  by  the 
equation  (11), 

a«2  =  (bnL  -  r)  (bnC  -  g). 

For  convenience  the  index  n  may  be  dropped  in  the  investiga- 
tion of  a  group  of  terms  of  current  and  voltage,  thus : 


a2  =  (bL  -  r)  (bC  -  g), 
and  the  following  substitutions  may  be  made : 

a  = 


a  =  h   +  jk, 
al  =  hl  +  jkv 
b  =  p  +  jq, 
from  which 

h  =  h}  VLC  and 
k  =  k,  VLC. 

Substituting  (18)  in  (16), 

(h,  +  jk,)2  =  \(p  +  jq)  -  y]  \(p  +  jq)  -  £l. 

L/J  oj 


(16) 

(17) 
(18) 


(19) 


(20) 


Carrying  out  and  separating  the  real  and  the  imaginary  terms, 
equation  (20)  resolves  into  the  two  equations  thus : 


and 


Substituting 


-.*(»".-£-$ 


(21) 


-C--A 

2VL     C/' 


(22) 
(23) 


424 

and 

into  (21)  gives 

or 
and 

or 


TRANSIENT  PHENOMENA 

p  =  S  +  u 

n     u*       —     O/Y 

riiKi      6*/> 

s2  -  (f  =  h*  —  k?  +  m2, 
sg  =  /i1A:1. 


| 
J 


(24) 
(25) 

(26) 


Adding  four  times  the  square  of  the  second  equation  to  the  square 
of  the  first  equation  of  (25)  and  (26)  respectively,  gives 


2   -  (f   -  7ft2)2   +  4  «V 

=  V(s2  +  cf  -  m2)2  +  4  £2m2 


(27) 


and 


+  q2    =       (h*  -  k*  +  m2)2  +  4  V&i2 


x2  +  A?!2  f  m2)2    -  4  A;,2™2 


and  substituting  (19),  gives,  by  (25),  (26)  and  (27),  (28) 


(28) 


k  = 


?  =  V(si  +  <f  -  TO2)2  +  4  <fm\ 


(29) 


or 


V(h*  +  It?  +  LCntf  --4 


(30) 


GENERAL   EQUATIONS 


425 


If,  however  (+  h  +  jk)  and  (u  +  s  +  jq)  satisfy  equation  (16), 
then  any  other  one  of  the  expressions 

(±  h  ±  jk)  and  (u  ±  s  ±  jq) 

also  satisfies  equation  (16),  providing  also  the  second  equation  of 
(25)  or  (26)  is  satisfied, 

hk  = 


sq] 


(31) 


that  is,  if  s  and  q  have  the  same  sign,  h  and  k  must  have  the  same 
sign,  and  inversely,  if  s  and  q  have  opposite  signs,  h  and  k  must 
have  opposite  signs. 

This  then  gives  the  corresponding  values  of  a  and  b  : 


(1)  a  =  +  h  +  jk 

+  h  -  jk 

(2)  a  =     -h  -  jk 

-  h  +  jk 

(3)  a  =     -  h  +  jk 

-  h  -  jk 

(4)  a  =  •  +  h  -  jk 

+  h  +  jk 


b  =  u  —  s  —  jq 

u  —  s  +  jq 

b  =  u  —  s  —  jq 

u  -  s  +  jq 

b  =  u  +  s  —  jq 

u  +  s  +  jq 

b  =  u  +  s  —  jq 

u  4-  s  +  ia 


(32) 


or  eight  pairs  of  corresponding  values  of  a  and  b. 

5.   Substituting  the  values   (1)   of    (32)  into   one  group  of 
terms  of  equations  (9)  and  (10), 

i  =  Ae-al~bi 
and 


bL  -r 


-al-bt 


a 


(33) 


gives 


4-  A  '£-(*- 


and  substituting  for  the  exponential  functions  with  imaginary 
exponents  their  trigonometric  expressions  by  the  equation 


cos  x 


426 

gives 


TRANSIENT  PHENOMENA 


il  =  s   '     (u   s)t\A1  [cos  (qt  —  kl)  +  j  sin  (qt  - 
+  A  /  [cos  (qt  —  kl)  —  j  sin  (qt  - 
=€-w-(M-a)<|  (A^  +  AI)  cos  (qt  —  kl)+j  (Al  —  A/)  sin  (g£  — , 

hence,  Ax  and  A/  must  be  conjugate  complex  imaginary  quan- 
tities, and  writing 

C,  =  A,  +  A/ 
and 

C/  =  j  (^  -  A/) 
gives 

Substituting  in  the  same  manner  in  the  equation  of  e,  in  (33), 
gives 

(u  —  s  —  jq)  L  —  r 


(34) 


jk 


-  j  A; 


-M-^-rtt  ([^  -s-jq)L-  r]  (h  -  jk)        +i(at_kl) 

.      (  h2  +  k2 

[(u-8  +  jq)  L-r](h  +  jk)      ,,„_ 


l 


hence  expanding,  and  substituting  the  trigonometric  expressions 

f(u  —  s)  L  —  r]  h  — 
h?  +  k* 

,  —  s}L  —  r\h  —  qkL 


_  .[(M-S)  L-r]  k  +  qhL\ 
tf  +  k*  I 


.[(u-s)L-r]  k  +  qkL 


A  /[cos  (qt  -  kl)  -  j  sin  (qt  -  kl)]l ,    (36) 


and  introducing  the  denotations 

=  qkL  +  h  [r  -  (u  -  s)  L]  _  qk  +  h  (m  +  s) 

h2  +  k2  h2  -f  k2 

,  =  k  [r  -  (u  -  s)  L]  -  ghL  _  k  (m  +  s)  -  qh 
°1=  h2  +  tf  h2  -f  k2 


(37) 


GENERAL  EQUATIONS 


427 


and  substituting  (37)  in  (36),  gives 
ei  =  £-hl-(u-s)t{(-cl  +  jc/)  AJcos  ($  -/c/)  +  /sin 

+  (  -  ct  -  jc/)  A/  [cos  ($  -  &)  -  j  sin  ($  -  kl}}  } 
=  €-«-(u-.)*{[_  Cj  (4i  +  AI/)+  jCi'(A1  -  A/)]  cos  (qt  -  kl) 
+  [-  /c^-  A/)-  c/(At+  A/)]sin(^  -  kl)\.    (38) 
Substituting  the  denotations  (34)  into  (38)  gives 
e,  =  €-"-<«-•>'{  (c/C/  -  C/7J  cos  (qt-  kl) 

-(c/Y  +  c/CJ  sin  (0  -A;/)}.  (39) 

The  second  group  of  values  of  a  and  b  in  equation  (32)  differs 
from  the  first  one  merely  by  the  reversal  of  the  signs  of  h  and  k, 
and  the  values  i2  and  e2  thus  are  derived  from  those  of  ^  and  el 
by  reversing  the  signs  of  h  and  k. 

Leaving  then  the  same  denotations  ct  and  c/  would  reverse 
the  sign  of  e2,  or,  by  reversing  the  sign  of  the  integration  con- 
stants C,  that  is,  substituting 

C2=-  (A,  +  A,') 
and  (40) 


,       } 

~~    ^-2  />  ^ 


the  sign  of  ^2  reverses;  that  is, 
ta  -  -  £+w-(w-s)/{C2  cos  ( 
and 


kl)  +C2'  sin 


(41) 


cos 


(42) 

The  third  group  in  equation  (32)  differs  from  the  first  one  by 
the  reversal  of  the  signs  of  h  and  s,  and  its  values  i3  and  e3  there- 
fore are  derived  from  il  and  el  by  reversing  the  signs  of  h  and  s. 

Introducing  the  denotations 

qk  -  h  (m  - 


c,  = 


h2  +  k2 

k  (m  —  s)  +  qh 
h2  +  tf 


L, 


and 


(43) 


Ct' -  j  (At- 


(44) 


TRANSIENT  PHENOMENA 


i3  =  fi +«-<«+«><  {C3  cos  (qt  -  kl)  +  C3'  sin  (qt  -  kl) }      (45) 


428 
gives 

and 

e3  =  e +*-<«+«><  { (ca'Cs'  -  c2C3)  cos  (qt  -  kl) 

~  (c2C8'  +  c2'C3)  sin  (qt  -  kl)}.  (46) 

The  fourth  group  in  (32)  follows  from  the  third  group  by  the 
reversal  of  the  signs  h  and  k}  and  retaining  the  denotations  c2 
and  c2';  but  introducing  the  integration  constants, 

C4  =  -  (A.  +  A/)    } 
and  L  (47) 

gives 

i4  =    -  e~hl-(u+s}i  {C4  cos  (qt  +  kl)  +  Ct'  sin  (qt  +  kl)}    (48) 
and 


e   = 


cos 


-  (c2C4'  +  ca7C4)  sin  (^  + 


(49) 


6.  This  then  gives  as  the  general  expression  of  the  equations  of 
the  electric  circuit: 


cos 


sn       _ 


sn 


and 


cos  (qt  -  kl)  +  C/  sin  (qt  -  kl)  }  (i3) 
cos  (^  +  kl)  +  C/  sin  (qt  +  M)  }]  (g 


i(ciGi   -  CjC/i)  cos  (gf  -  /c/) 

-  (c/Ct  +  c^/)  sin  (qt  -  kl) }  (e,) 

-  (Ci'C2  +  c^/)  sin  ($  +  A;Z) }  (e2) 

[c2'C3   —  c2C3)  cos  (qt  —  kl) 

'c2C4'  —  c2C4)  cos  (qt  +  A;Z) 


(50) 


(51) 


GENERAL  EQUATIONS 


429 


where  Cv  C/,  Cv  C{,  Cv  C,',  C4>  C/  and  two  of  the  four  values 
s,  g,  hj  k  are  integration  constants,  depending  on  the  terminal 
conditions,  and 

qk  +  h  (m  +  s)  _ 


,  _  k(m  +  s)  -  qh 
i   :          #  +  # 

<?&  —  /&  (m  —  s) 


(m  -  s)  +  g/t 


T 


and 


-.(£-  1) 


and  /i,  A;  and  s,  </  are  related  by  the  equations 

h  =  VLC  \/i  j/2,2  +  s2  -  (f  -  m2}, 

A;  = 
and 


m2}, 


R*  =  V(s2  +  q2  -  m2)2  +  4  <fm2\ 
hence,  h2  +  #  =  LC/2^, 

or 


s  = 


and 
hence, 


-  4 


(52) 


(53) 


(54) 


(55) 


(56) 


(57) 


430 
Writing 


TRANSIENT  PHENOMENA 


D  (qt  ±  kl)  =  C  cos  (qt  ±  kl)  +  C'  sin  (qt  ±  kl)         (58) 


and 


H  (qt  ±  kl)  =  (c'C"  -  cC)  cos  (qt  ±  kl) 

-  (c'O  +  cC')  sin  (qt  ±  kl), 


equations  (50)  and  (51)  can  be  written  thus: 


(qt-kl)  - 


and 


(59) 


(60) 


(ef) 


(61) 


CHAPTER  II. 

DISCUSSION  OF  GENERAL  EQUATIONS. 

7.  In  the  preceding  chapter  the  general  equations  of  current 
and  voltage  were  derived  for  a  circuit  or  section  of  a  circuit 
having  uniformly  distributed  and  constant  values  of  r,  L,  g,  C. 
These  equations  appear  as  a  sum  of  groups  of  four  terms  each, 
characterized  by  the  feature  that  the  four  terms  of  each  group 
have  the  same  values  of  s,  q,  h,  k. 

Of  the  four  terms  of  each  group,  iv  iv  i3,  i4  or  ev  ev  es,  e4 
respectively  (equations  (50)  and  (51)),  two  contain  the  angles 
(qt  —  kl):  iv  e1  and  iz,  e3;  and  two  contain  the  angles  (qt  +  kl): 
i2,  e2  and  i4,  e4. 

In  the  terms  iv  e^  and  iz,  e3,  the  speed  of  propagation  of  the 
phenomena  follows  from  the  equation 

qt  -  kl  =  constant, 

thus: 

ti  q 

dt   r.*1 

hence  is  positive,  that  is,  the  propagation  is  from  lower  to  higher 
values  of  I,  or  towards  increasing  I. 
In  the  terms  iv  e2  and  i4,  e4,  the  speed  of  propagation  from 

qt  +  kl  =  constant 
is 

dl_    _q 
Jt~    ~k 

hence  is  negative,  that  is,  the  propagation  is  from  higher  to 
lower  values  of  I,  or  towards  decreasing  I. 

Considering  therefore  iv  el  and  i3J  e3  as  direct  or  main 
waves,  iv  e2  and  i4,  e4  are  their  return  waves,  or  reflected  waves, 
and  iv  e2  is  the  reflected  wave  of  iv  e^  i4,  e4  is  the  reflected  wave 
of  iv  ey 

431 


432  TRANSIENT  PHENOMENA 

Obviously,  i2J  e2  and  i^  e±  may  be  considered  as  main  waves, 
and  then  iv  et  and  i3,  e3  are  reflected  waves.  Substituting  ( -  I) 
for  (+  I)  in  equations  (50)  and  (51),  that  is,  looking  at  the 
circuit  in  the  opposite  direction,  terms  i2,  e2  and  iv  e1  and  terms 
i4,  64  and  iv  e3  merely  change  places,  but  otherwise  the  equations 
remain  the  same,  except  that  the  sign  of  i  is  reversed,  that  is, 
the  current  is  now  considered  in  the  opposite  direction. 

Each  group  thus  consists  of  two  waves  and  their  reflected 
waves:  ^  -  i2  and  et  +  e2  is  the  first  wave  and  its  reflected 
wave,  and  i3  -  i4  and  e3  +  e4  is  the  second  wave  and  its 
reflected  wave. 

In  general,  each  wave  and  its  reflected  wave  may  be  con- 
sidered as  one  unit,  that  is,  we  can  say:  if  =  il  —  i2  and  e'  = 
e^  +  e2  is  the  first  wave,  and  i"  =  i3  —  i4  and  e"  =  e3  +  e4  is 
the  second  wave. 

In  the  first  wave,  i',  e',  the  amplitude  decreases  in  the  direction 
of  propagation,  e~wfor  rising,  e+hl  for  decreasing  I,  and  the 
wave  dies  out  with  increasing  time  t  by  e -<«-*>*  =  £~ut  £+stf 

In  the  second  wave,  i",  eft ',  the  amplitude  increases  in  the 
direction  of  propagation,  e+w  for  rising,  s~hl  for  decreasing  I, 
but  the  wave  dies  out  with  the  increasing  time  t  by  £-(tt+s>< 
=  s""*  e~st,  that  is,  faster  than  the  first  wave. 

If  the  amplitude  of  the  wave  remained  constant  throughout 
the  circuit  —  as  would  be  the  case  in  a  free  oscillation  of  the 
circuit,  in  which  the  stored  energy  of  the  circuit  is  dissipated, 
but  no  power  supplied  one  way  or  the  other  —  that  is,  if  h  =  0, 
from  equation  (56)  s  =  0;  that  is,  both  waves  coincide  and  form 
one,  which  dies  out  with  the  time  by  the  decrement  e~ut. 

It  thus  follows:  In  general,  two  waves,  with  their  reflected 
waves,  traverse  the  circuit,  of  which  the  one,  i",  e",  increases  in 
amplitude  in  the  direction  of  propagation,  but  dies  out  corre- 
spondingly more  rapidly  in  time,  that  is,  faster  than  a  wave  of 
constant  amplitude,  while  the  other,  i',  e' ,  decreases  in  amplitude 
but  lasts  a  longer  time,  that  is,  dies  out  slower  than  a  wave  of 
constant  amplitude.  In  the  one  wave,  i",  e" ',  a  decrease  of 
amplitude  takes  place  at  a  sacrifice  of  duration  in  time,  while  in 
the  other  wave,  i',  e',  a  slower  dying  out  of  the  wave  with  the 
time  is  produced  at  the  expense  of  a  decrease  of  amplitude  during 
its  propagation,  or,  in  i",  e"  duration  in  time  is  sacrificed  to 
duration  in  distance,  and  inversely  in  i',  e'. 


DISCUSSION  OF  GENERAL  EQUATIONS  433 

It  is  interesting  to  note  that  in  a  circuit  having  resistance, 
inductance,  and  capacity,  the  mathematical  expressions  of  the 
two  cases  of  energy  flow;  that  is,  the  gradual  or  exponential 
and  the  oscillatory  or  trigonometric,  are  both  special  cases  of 
the  equations  (60)  and  (61),  corresponding  respectively  to 
q  =  o,  k  =  0  and  to  h  =  0,  s  =  0. 

8.   In  the  equations  (50)  and  (51) 

qt  =  2x 
gives  the  time  of  a  complete  cycle,  that  is,  the  period  of  the  wave, 


and  the  frequency  of  the  wave  is 

/  =  -L 
2 

kl    =   27T 

gives  the  distance  of  a  complete  cycle,  that  is,  the  wave  length, 

W  7          7 

k 
(u  —  s)  t  =  1     and     (u  +  s)  t  =  1 

give  the  time, 

*/'-  —      and      t"=  -*—, 


during  which  the  wave  decreases  to  -  =  0.3679  of  its  value,  and 

hi  =  1 
gives  the  distance, 


over  which  the  wave  decreases  to  -  =  0.3679  of  its  value; 

£ 

that  is,  q  is  the  frequency  constant  of  the  wave, 

f    -  -       I  I:  «'—'••  (62) 

>      2V          °~' 


434  TRANSIENT  PHENOMENA 

k  is  the  wave  length  constant, 


(63) 


(u  -  s)  and  (u  -f  s)  are  the  time  attenuation  constants  of  the  wave, 

1 

) 

(64) 


U  +  S 


and  h  is  the  distance  attenuation  constant  of  the  wave, 

L -I. 


(65) 


9.  If  the  frequency  of  the  current  and  e.m.f.  is  very  high, 
thousands  of  cycles  and  more,  as  with  traveling  waves,  lightning 
disturbances,  high-frequency  oscillations,  etc.,  q  is  a  very  large 
quantity  compared  with  s,  u,  m,  h,  k,  and  k  is  a  large  quantity 
compared  with  h,  then  by  dropping  in  equations  (50)  to  (61)  the 
terms  of  secondary  order  the  equations  can  be  simplified. 

From  (54), 


^  =  V(s2  +  q2  -  m2)2  +  4  q2m2  =  V(q2  +  m2)2  +  2  s2  (q2  -  m2) 


=  q2  +  m2  +  s2 
=  q2  -f  m2  +  s2 


-m 


(f 


-  q  -m     = 

+rn2}  =VLO(q2+  m2)=qVLO, 
h2  +  tf  =  (s2  +  q2)  LC  =  q2LC, 


•j 


(66) 


and 


+  h  (m  +  s)      _qL 


DISCUSSION  OF  GENERAL  EQUATIONS 


435 


qk  —  h  (m  —  s)          qL         IL 
=         A'  +  yfc2     ~L="T      \C' 


,  _  k  (m  +  s)  -  qh  J  _  q  VLC  (m  +  s)  -  qs  VLC 

q2LC 


r  _m    /L 
~V' 


f  __  k  (m  -  s)  +  qh         q  VTC  (m  -  s)  +  qs  VTC 


h2  +  k2 


that  is, 


and 


Writing 


q2LC 

\/i 


Cl    -     C2  q 


<r  = 


(67) 


(68) 


where  <r  is  the  reciprocal  of  the  frequency  of  propagation  (velocity 
of  light),  we  have 

h  =  o-s, 

k  =  o-q, 


(69) 


and 


m 
—  c 

q 


(70) 


and  introducing  the  new  independent  variable,  as  distance, 


we  have 
and 


hi  =  si;. 


(71) 
(72) 


436  TRANSIENT  PHENOMENA 

hence,  the  wave  length  is  given  by 

qX  =  2n 
as 

V-^j  (73) 

and  since  the  period  is 


it  follows  that  by  the  introduction  of  the  denotation  (71)  dis- 
tances are  measured  with  the  velocity  of  propagation  as  unit 
length,  and  wave  length  /„,  and  period  t0  thus  have  the  same 
numerical  values. 
Substituting  now  in  equations  (50)  and  (51)  gives 


D2[q  (t+X)]  (if) 
D4[q  (t  +  X)]}(iff)  (74) 
and 
e  =  e-e+'^ffjg  (t-X)]  +  e+s«+»  H2[q(t  +  X)]  (e') 


where 

£  fe  (t  ±  *)]  =  C  cos  q  (t  ±  X)  +  C'  sin  q  (t  ±  X) 
and 

H[q  (t  ±  /I)]  -  \-C'-  C^q  (t  ±  X) 


ng(/  ±  >l).  (76) 

10.  As  seen  from  equations  (74)  and  (75),  the  waves  are 
products  of  £~ut  and  a  function  of  (t  -  X)  for  the  main  wave, 
(t  +  X)  for  the  reflected  wave,  thus  : 

*\  +  *'.  =  *~utL  (t  -  V  } 

and  (77) 

i2  +  i4  =  e~utf2  (t  +  A);J 


DISCUSSION  OF  GENERAL  EQUATIONS  437 

hence,  for  constant  (t  -  X)  on  the  main  waves,  and  for  constant 
(t  +  X)  on  the  reflected  waves,  we  have 


and  (78) 


that  is,  during  its  passage  along  the  circuit  the  wave  decreases 
by  the  decrement  e~ut,  or  at  a  constant  rate,  independent  of 
frequency,  wave  length,  etc.,  and  depending  merely  on  the 
circuit  constants  r,  L,  g,  C.  The  decrement  of  the  traveling 
wave  in  the  direction  of  its  motion  is 


and  therefore  is  independent  of  the  character  of  the  wave,  for 
instance  its  frequency,  etc. 

11.  The  physical  meaning  of  the  two  waves  i'  and  e'  can  best 
be  appreciated  by  observing  the  effect  of  the  wave  when  travers- 
ing a  fixed  point  X  of  the  circuit. 

Consider  as  example  the  main  wave  only,  i'  =  i^  +  ia,  and 
neglect  the  reflected  waves,  for  which  the  same  applies. 

From  equation  (74), 

i  =  e  —*-("-*>«  DJg  (t  -  ;)]  +  e+s*-(u+s»D3[q(t  -  /I)];  (79) 
or  the  absolute  value  is 


(80) 

where  Dt  and  D3  have  to  be  combined  vectorially. 

Assuming  then  that  at  the  time  t  =  0,  7  =  0,  for  constant  X 
we  have 

/  =  D  (£-<«-•>«  -  fi  -<"+«>«),  (81) 

the  amplitude  of  7  at  point  L 

Since  (81)  is  the  difference  of  two  exponential  functions  of 
different  decrement,  it  follows  that  as  function  of  the  time  t,  I 
rises  from  0  to  a  maximum  and  then  decreases  again  to  zero,  as 
shown  in  Fig.  98,  where 


73  -  De  - 

r-  £  -  1* 

and  the  actual  current  i  is  the  oscillatory  wave  with  7  as  envelope. 


438 


TRANSIENT  PHENOMENA 


The  combination  of  two  waves  thus  represents  the  passage  of 
a  wave  across  a  given  point,  the  amplitude  rising  during  the 
arrival  and  decreasing  again  after  the  passage  of  the  wave. 


Fig.  98.    Amplitude  of  electric  traveling  wave. 

12.  If  h  and  so  also  s  equal  zero,  i' ,  er  and  i" ',  e"  coincide  in 
equations  (74)  and  (75),  and  Cl  and  C3  thus  can  be  combined 
into  one  constant  Bv  C2  and  C4  into  one  constant  J52,  thus : 


C3  =  Bv 
Ct  -  Bv 
Cs'  =  Bt', 


(82) 


and  (74),  (75)  then  assume  the  form 

i  =  s-ut  ^  j[JB1  cos  q(t-  X)  +  £/  sin  q(t  -  X)] 
-  [B2  cos  q(t  +  *)  +  B2'  sin  q  (t  +  A)]  }  , 


(83) 


(84) 

These  equations  contain  the  distance  X  only  in  the  trigono- 
metric but  not  in  the  exponential  function;  that  is,  i  and  e 
vary  in  phase  throughout  the  circuit,  but  not  in  amplitude;  or, 
in  other  words,  the  oscillation  is  of  uniform  intensity  throughout 
the  circuit,  dying  out  uniformly  with  the  time  from  an  initial 
maximum  value;  however,  the  wave  does  not  travel  along  the 


DISCUSSION  OF  GENERAL  EQUATIONS  439 

circuit,  but  is  a  stationary  or  standing  wave.  It  is  an  oscillatory 
discharge  of  a  circuit  containing  a  distributed  r,  L,  g,  C,  and 
therefore  is  analogous  to  the  oscillating  condenser  discharge 
through  an  inductive  circuit,  except  that,  due  to  the  distributed 
capacity,  the  phase  changes  along  the  circuit.  The  free  oscilla- 
tions of  a  circuit  such  as  a  transmission  line  are  of  this  character. 
For  A  =  0,  that  is,  assuming  the  wave  length  of  the  oscillation 
as  so  great,  hence  the  circuit  as  such  a  small  fraction  of  the  wave 
length,  that  the  phase  of  i  and  e  can  be  assumed  as  uniform 
throughout  the  circuit,  the  equations  (83)  and  (84)  assume  the 
form 

i  =  £-^{B0  cos  qt  +  BQ'  sin  qt\ 
and  (85) 


these  are  the  usual  equations  of  the  condenser  discharge  through 
an  inductive  circuit,  which  here  appear  as  a  special  case  of  a 
special  case  of  the  general  circuit  equations. 

If  q  equals  zero,  the  functions  D  and  H  in  equations  (74)  and 
(75)  become  constant,  and  these  equations  so  assume  the  form 


and 


e  = 


(86) 


where 

B=^0'-C.  (87) 

This  gives  expressions  of  current  and  e.m.f.  which  are  no 
longer  oscillatory  but  exponential,  thus  representing  a  gradual 
change  of  i  and  e  as  functions  of  time  and  distance,  corresponding 
to  the  gradual  or  logarithmic  condenser  discharge.  For  A  =  0, 
these  equations  change  to  the  equations  of  the  logarithmic  con- 
denser discharge. 


440 


TRANSIENT  PHENOMENA 


These  equations  (86)  are  only  approximate,  however,  since  in 
them  the  quantities  s,  u,  h  have  been  neglected  compared  with 
q,  assuming  the  latter  as  very  large,  while  now  it  is  assumed  as 
zero. 

13.  If,  however, 


that  is, 


or 


L     C 


r  -H  g  =  L  H-  C, 


(89) 


or,  in  words,  the  power  coefficients  of  the  circuit  are  proportional 
to  the  energy  storage  coefficients,  or  the  time  constant  of  the 

electromagnetic  field  of  the  circuit,  —  ,  equals  the  time  constant 

L 

of  the  electrostatic  field  of  the  circuit,  -^ ,  then 


u  =  —  =  —  =  time  constant  of  the  circuit,  (90) 

L      C 


and  from  equation  (54) 

R*  =  s2  +  (f, 
h  =  VWs  =  as, 
k  =  VWq  =  aq, 

and  from  equation  (52) 

=  L      .  /L 

c/  m  0, 


(91) 


and 


/  =  0; 


(92) 


DISCUSSION  OF  GENERAL  EQUATIONS  441 

hence,  substituting  in  equations  (50)  and  (51), 

e     L     (_+s(£  — A)   T~)    r~  /*          i\"i  +  s  (t  +  A)  T~)    r~  /*     i      J\T 

I  *  i  Li  \  /  J  "^2  LI  \      •       /  J 

+  c-'<'-A>  D.b  (f  -  X)]  -  £-'«+*>  DJg  (t  +  i)]}     (93) 
and 


+  *~J    A)z)3b(^->i)]  +  £-<    '>D4fe(r-f 

(94) 

These  equations  are  similar  to  (74)  and  (75),  but  are  derived 
here  for  the  case  m  =  0,  without  assumptions  regarding  the 
relative  magnitude  of  #and  the  other  quant  it  ies :  "  distortionless 
circuit." 

These  equations  (93)  and  (94)  therefore  also  apply  for  q  =  0, 
and  then  assume  the  form 

S^-A)]-[C2£+S('+A)  +  C4£-S(^A)]},(95) 


e=- 

+  [C2e+8«+»  +  C4£-S(<+A)]}.  (96) 

These  equations  (95)  and  (96)  are  the  same  as  (86),  but  in  the 
present  case,  where  m  =  0,  apply  irrespective  of  the  relative 
values  of  the  quantities  s,  etc. 

Therefore  in  a  circuit  in  which  m  =  0  a  transient  term  may 
appear  which  is  not  oscillatory  in  time  nor  in  space,  but 
changing  gradually. 

If  the  constant  h  in  equations  (50)  and  (51)  differs  from  zero, 
the  oscillation  (using  the  term  oscillation  here  in  the  most  general 
sense,  that  is,  including  also  alternation,  as  an  oscillation  of  zero 
attenuation)  travels  along  the  circuit,  but  it  becomes  stationary, 
as  a  standing  wave,  for  h  =  0;  that  is,  the  distance  attenuation 
constant  h  may  also  be  called  the  propagation  constant  of  the 
wave. 

h  =  0  thus  represents  a  wave  which  does  not  propagate  or 
move  along  the  circuit,  but  stands  still,  that  is,  a  stationary  or 
standing  wave. 


CHAPTER  III. 

STANDING  WAVES. 

14.   If  the  propagation  constant  of  the  wave  vanishes, 

h  =  0, 

the  wave  becomes  a  stationary  or  standing  wave,  and  the  equa- 
tions of  the  standing  wave  are  thus  derived  from  the  general 
equations  (50)  to  (61),  by  substituting  therein  h  =  0,  which 
gives 

R2  =  V(k2  -  LCm2)2;  (97) 

hence,  if  k2  >  LCm2, 

R2  =  tf-  LCm2; 

and  if  /c2  <  LCm2, 

R2  =  LCm2'-  tf. 

Therefore,  two  different  cases  exist,  depending  upon  the  rela- 
tive values  of  Ar*  and  LCm2,  and  in  addition  thereto  the  inter- 
mediary or  critical  case,  in  which  k2  =  LCm2. 

These  three  cases  require  separate  consideration. 


is  a  circuit  constant,  while  k  is  the  wave  length  constant,  that  is, 
the  higher  k  the  shorter  the  wave  length. 
A.  Short  waves, 

k2  >  LCm2,  (99) 

hence, 

R2  =  k2  -  LCm2  (100) 

and 


q  =  V  ^  -  ™\ 

442 


STANDING   WAVES 


443 


or  approximately,  for  very  large  k, 


Herefrom  then  follows 


and 


VLC 


°l=  k     c' 

mL 
c,   -T-c, 

-1± - 

2"  k 

c'=---c' 

2     k 


(102) 


(103) 


Substituting  now  h  =  0  and  (101),  (103)  in  equations  (50), 
(51),  the  two  waves  i',  e'  and  i",  e"  coincide,  and  all  the  expo- 
nential terms  reduce  to  e~ut',  hence,  substituting 


and 
gives 


,  +  Cv 
2  +  C4, 

i  +  C*', 
2f  +  €4, 


(104) 


= 


t  {[B^  cos  (qt  -  kl)  +  BS  sin  (qt  -  kl)] 

-  [B2  cos  (qt  +  kl)  +  B2'  sin  (qt  +  kl)]}      (105) 


and 


L 

C  =  k 


f-qBJ  cos  (qt-kl)-(mB,  +  qB,'}  sin  (qt-kl)] 
+  [(mB2'-qB3)  cos  (qt  +  Jd)-(mB2  +  qBJ)  sin  (qt  +  kl)]}.  (106) 


Equations  (105)  and  (106)  represent  a  stationary  electrical  oscil- 
lation or  standing  wave  on  the  circuit. 


B.   Long  waves, 


k2  <  LCm2] 


(107) 


444 

hence, 

and 


TRANSIENT  PHENOMENA 


R22  =  LCm2  -  k2, 


s  = 


(108) 


(109) 


or  approximately,  for  very  small  values  of  &, 

1    r 


herefrom  then  follows 


(HO) 


ci  =  c2  =  0, 


and 


(m  +  s)  L 
~T~ 


(m  —  s)  L 


(111) 


Substituting  now  h  =  Oand  (109),  (111)  into  (50)  and  (51),  the 
two  waves  H ',  e'  and  i",  e"  remain  separate,  having  different  expo- 
nential terms,  e~(u-s»  and  t^"**  but  in  each  of  the  two  waves 
the  main  wave  and  the  reflected  wave  coincide,  due  to  the 
vanishing  of  q. 

Substituting  then 


=  C-  C 


and 
gives 


(112) 


sinkl] 
(113) 


STANDING  WAVES 


445 


and 


*{[(*»  +  s)  £/£+s<  +  (m  -  s)  £2'£~8<]  cos  W 
+  [(m  +  s)  B1£+rf  +  (m  -  s)  B2e~st]  sin  kl} 


s')sinA;Z] 


(114) 


+  (BlS  +  «  -  B2e~st)smkl]} 

Equations  (113)  and  (114)  represent  a  gradual  or  exponential 
circuit  discharge,  and  the  distribution  still  is  a  trigonometric 
function  of  the  distance,  that  is,  ^  wave  distribution,  but  dies  out 
gradually  with  the  time,  without  oscillation. 

C.  Critical  case, 


hence, 


o, 


=  0, 


(115) 


(116) 


and 


c2  =  0, 
raL 


(117) 


and  all  the  main  waves  and  their  reflected  waves  coincide  when 
substituting  h  =  0,  (116),  (117)  in  (50)  and  (51). 
Hence,  writing 


and 
gives 


B   =  C,   -  C2  +  C3  -  C,    1 
B'  =  CY  4-  C2'  +  C,7  +  C/    J 

i  =  fi-"1  {B  cos  kl  -  B'  sin  Id] 


(118) 


(119) 


446  TRANSIENT  PHENOMENA 

and 

e  =  y  — £-M<  {5'  cos  kl  +  B  sin 

In  the  critical  case,  (119)  and  (120),  the  wave  is  distributed 
as  a  trigonometric  function  of  the  distance,  but  dies  out  as  a 
simple  exponential  function  of  the  time. 

15.  An  electrical  standing  wave  thus  can  have  two  different 
forms:  it  can  be  either  oscillatory  in  time  or  exponential  in  time, 
that  is,  gradually  changing.  It  is  interesting  to  investigate  the 
conditions  under  which  these  two  different  cases  occur. 

The  transition  from  gradual  to  oscillatory  takes  place  at 

k*  =  m2LC;  (121) 

for  larger  values  of  k  the  phenomenon  is  oscillatory;  for  smaller, 
exponential  or  gradual. 

Since  k  is  the  wave  length  constant,  the  wave  length,  at  which 
the  phenomenon  ceases  to  be  oscillatory  in  time  and  becomes  a 
gradual  dying  out,  is  given  by  (63)  as 

27T 

2,  (122) 


m 


Vie 


In  an  undamped  wave,  that  is,  in  a  circuit  of  zero  r  and  zero  g, 
in  which  no  energy  losses  occur,  the  speed  of  propagation  is 


and  if  the  medium  has  unit  permeability  and  unit  inductivity,  it 
is  the  speed  of  light, 

S0  =  3  X  1010.  (124) 

In  an  undamped  circuit,  this  wave  length  lWo  would  correspond 
to  the  frequency, 


STANDING   WAVES  447 

hence,  from  (62), 

'•-,Vr.-  <125> 

The  frequency  at  the  wave  length  lWo  is  zero,  since  at 
this  wave  length  the  phenomenon  ceases  to  be  oscillatory  ;  that  is, 
due  to  the  energy  losses  in  the  circuit,  by  the  effective  resistance  r 
and  effective  conductance  g,  the  frequency  /  of  the  wave  is 
reduced  below  the  value  corresponding  to  the  wave  length  lw, 
the  more,  the  greater  the  wave  length,  until  at  the  wave  length 
lWo  the  frequency  becomes  zero  and  the  phenomenon  thereby 
non-oscillatory.  This  means  that  with  increasing  wave  length 
the  velocity  of  propagation  of  the  phenomenon  decreases,  and 
becomes  zero  at  wave  length  lWo. 

If  m2LC  =  0, 

k0  =  0  and  lWo  =  oo  ; 

that  is,  the  standing  wave  is  always  oscillatory. 

If  m?LC  =  oo, 

k0  =  oo  and  lWo  =  0; 

that  is,  the  standing  wave  is  always  non-oscillatory,  or  gradually 
dying  out. 

In  the  former  case,  m?LC  =  0,  or  oscillatory  phenomenon, 
substituting  for  m2,  we  have 


and 

r  _L 

g    c' 

or 

rC  —  gL  =  0    (distortionless  circuit). 

In  the  latter  case,  m2LC  =  oo  ,  or  non-oscillatory  or  exponen 
tial  standing  wave,  we  have 


r  \  —  —  Q\  —  =00 


448  TRANSIENT  PHENOMENA 

and  since  neither  r,  g,  L,  nor  C  can  be  equal  infinity  it  fol- 
lows that  either  L  =  0  or  C  =  0. 

Therefore,  the  standing  wave  in  a  circuit  is  always  oscillatory, 
regardless  of  its  wave  length,  if 

rC  -  gL  =  0,  (126) 

or 

-  =  §J  (127) 

that  is,  the  ratio  of  the  energy  coefficients  is  equal  to  the  ratio 
of  the  reactive  coefficients  of  the  circuit. 

The  standing  wave  can  never  be  oscillatory,  but  is  always 
exponential,  or  gradually  dying  out,  if  either  the  inductance  L  or 
the  capacity  0  vanishes ;  that  is,  the  circuit  contains  no  capacity 
or  contains  no  inductance. 

In  all  other  cases  the  standing  wave  is  oscillatory  for  waves 

shorter  than  the  critical  value  L   =  -— ,  where 

0       V 

-  9  V  §}  >  (128) 

and  is  exponential  or  gradual  for  standing  waves  longer  than  the 
critical  wave  length  lWo;  or  for  k  <  ko  the  standing  wave  is 
exponential,  for  k  >  ka  it  is  oscillatory.0 

The  value  kQ  =  m  VLC  thus  takes  a  similar  part  in  the  theory 
of  standing  waves  as  the  value  r02  =  4  L0C0  in  the  condenser 
discharge  through  an  inductive  circuit;  that  is,  it  separates 
the  exponential  or  gradual  from  trigonometric  or  oscillatory 
conditions. 

The  difference  is  that  the  condenser  discharge  through  an 
inductive  circuit  is  gradual,  or  oscillatory,  depending  on  the 
circuit  constants,  while  in  a  general  circuit,  with  the  same  circuit 
constants,  usually  gradual  as  well  as  oscillatory  standing  waves 
exist,  the  former  with  greater  wave  length,  or 

m  VLC  >  k,  (129) 

the  latter  with  shorter  wave  length,  or 

m  VLC  <  k.  (130) 


STANDING   WAVES  449 

An  idea  of  the  quantity  k0,  and  therewith  the  wave  length  lu,o, 
at  which  the  frequency  of  the  standing  wave  becomes  zero,  or 
the  wave  non-oscillatory,  and  of  the  frequency  /0,  which,  in  an 
undamped  circuit,  will  correspond  to  this  critical  wave  length  lWo, 
can  best  be  derived  by  considering  some  representative  numerical 
examples. 

As  such  may  be  considered: 

(1)  A  high-power  high-potential  overhead  transmission  line. 

(2)  A  high-potential  underground  power  cable. 

(3)  A  submarine  telegraph  cable. 

(4)  A  long-distance  overhead  telephone  circuit. 

(1)  High-power  high-potential  overhead  transmission  line. 

16.  Assume  energy  to  be  transmitted  120  miles,  at  40,000 
volts  between  line  and  ground,  by  a  three-phase  system  with 
grounded  neutral.  The  line  consists  of  copper  conductors,  wire 
No.  00  B.  and  S.  gage,  with  5  feet  between  conductors. 

Choosing  the  mile  as  unit  length, 

r  =  0.41  ohm  per  mile. 

The  inductance  of  a  conductor  is  given  by 


=  I  (2  loge  lf 


10~9,  in  henrys,  (131) 


where  I  =  the  length  of  conductor,  in  cm.;  lr  =  the  radius  of 
conductor;  ld  =  the  distance  from  return  conductor,  and  /*  = 
the  permeability  of  conductor  material.  For  copper,  fi  =  1. 

As  one  mile  equals  1.61  X  105  cm.,  substituting  this,  and 
reducing  the  natural  logarithm  to  the  common  logarithm,  by  the 
factor  2.3026,  gives 

L  =  f  0.7415  log  ^  +  0.0805\  in  mh.  per  mile.         (132) 

,  For  lr  =  0.1825  inch  and  ld  =  60  inches, 

L  =  1.95  mh.  per  mile. 

The  capacity  of  a  conductor  is  given  by 

C  =  I  I  -          *—=\  109,  in  farads,  (133) 


450  TRANSIENT  PHENOMENA 

where  S0  =  3  X  1010  =  the  speed  of  light,  and  d  =  the  allow- 
ance for  capacity  of  insulation,  tie  wires,  supports,  etc.,  assumed 
as  5  per  cent. 

Substituting  £0,  and  reducing  to  one  mile  and  common  loga- 
rithm, gives 


mf.;  (134) 

logf 

lr 

hence,  in  this  instance, 


C  =  0.0162  mf. 

Estimating  the  loss  in  the  static  field  of  the  line  as  400  watts 
per  mile  of  conductor  gives  an  effective  conductance, 


which  gives  the  line  constants  per  mile  as  r  =  0.41  ohm;  L  = 
1.95X10-3  henry;  g  =  0.25  X  10~6  mho,  and  C  =  0.0162  X  lO"6 
farad. 
Herefrom  then  follows 


:>-i.S-.S-'* 

a-  =  VLC  =  V31.6  X  10~6  =  5.62  X  lO"6, 
&0  =  ra\/57  =  545  X  10~6; 
hence,  the  critical  wave  length  is 

^o  =:  IT  =  11»500  miles, 
/c0 

and  in  an  undamped  circuit  this  wave  length  would  correspond 
to  the  frequency  of  oscillation, 

-       m 
/0=  ^~  =  15.7  cycles  per  See. 


STANDING   WAVES  451 

Since  the  shortest  wave  at  which  the  phenomenon  ceases  to  be 
oscillatory  is  11,500  miles  in  length,  and  the  longest  wave  which 
can  originate  in  the  circuit  is  four  times  the  length  of  the  circuit, 
or  480  miles,  it  follows  that  whatever  waves  may  originate  in  this 
circuit  are  by  necessity  oscillatory,  and  non-oscillatory  currents 
or  voltages  can  exist  in  this  circuit  only  when  impressed  upon  it 
by  some  outside  source,  and  then  are  of  such  great  wave  length 
that  the  circuit  is  only  an  insignificant  fraction  of  the  wave,  and 
great  differences  of  voltage  and  current  of  non-oscillatory  nature 
cannot  exist. 

Since  the  difference  in  length  between  the  shortest  non- 
oscillatory  wave  and  the  longest  wave  which  can  originate  in  the 
circuit  is  so  very  great,  it  follows  that  in  high-potential  long- 
distance transmission  circuits  all  phenomena- which  may  result 
in  considerable  potential  differences  and  differences  of  current 
throughout  the  circuit  are  oscillatory  in  nature,  and  the  solution 
case  (A)  is  the  one  the  study  of  which  is  of  the  greatest 
importance  in  long-distance  transmissions. 

With  a  length  of  circuit  of  120  miles,  the  longest  standing  wave 
which  can  originate  in  the  circuit  has  the  wave  length 

lw  =  480  miles, 


and  herefrom  follows 


k  =     -  =  0.0134 


and 

ft  0.01342 


LC      31.6  X  10-12 
hence,  in  the  expression  of  q  in  equation  (101), 


=  V  5.7  X  106  -  0.00941  X  10e, 

Ar5 
m2  is  negligible  compared  with  —  ;  that  is, 


452  TRANSIENT  PHENOMENA 

or 

/  =  —  =  380  cycles  per  sec. 

2  7T 

Hence,  even  for  the  longest  standing  wave  which  may  origi- 
nate in  this  transmission  line,  q  =  2380  is  such  a  large  quantity 
compared  with  m  =*  97  that  m  can  be  neglected  compared  with 
q,  and  for  shorter  waves,  the  overtones  of  the  fundamental  wave, 
this  is  still  more  the  case;  that  is,  in  equations  (105)  and  (106) 

the  terms  with  m  may  be  dropped.     In  equation  (106)  ~  thus 

fc 

become  common  factors,  and  since  from  equation  (135) 


by  substituting  m  =  0  and  (136)  in  (105)  and  (106)  we  get  the 
general  equations  of  standing  waves  in  long-distance  transmission 


lines,  thus  : 


i  =  e-*  {[Bl  cos  (qt  -  kl)  +  £/  sin  (qt  -  kl)] 

-  [B2  cos  (qt  +  kl)  +.  £2'  sin  (qt  +  kl)]},  (137) 


=  -  \J  p£~ut{  [#i  cos  (qt  -  kl)  +  £/  sin  (qt  -  kl)] 

+  [£2cos  (qt  +  kl)  +  £2'sin  (qt  +  kl)]},  (138) 


or 


e  =  e-«rf{  [Al  cos  (qt  +  kl)  +  A/  sin  (qt  +  kl)] 

+  [A2  cos  (qt  -  kl)  +  A2'  sin  (qt  -  kl)]},  (139) 

fc 

i  =  y  ^  £~ui{[A,  cos  (qt  +  kl)  +  A/  sin  (qt  +  kl)] 

-  [A2  cos  (qt  -  kl)  +  A2'  sin  (qt  -  kl)]},  (140) 
where 


2>        A'  •    -  V      5/,  etc. 

(2)  High-potential  underground  power  cable. 
17.   Choose  as   example   an  underground  power   cable  of  20 
miles  length,  transmitting  energy  at  7000  volts  between  con- 


STANDING  WAVES  453 

ductor  and  ground  or  cable  armor,  that  is,  a  three-phase  three- 
conductor  12,000-volt  cable. 

Assume  the  conductor  as  stranded  and  of  a  section  equiva- 
lent to  No.  00  B.  and  S.  G. 

Calculating  the  constants  in  the  same  manner,  except  that 
the  expression  for  the  capacity,  equation  (119),  multiplies  with 
the  dielectric  constant  or  specific  capacity  of  the  cable  insula- 
tion, and  that  f  ig  verv  small,  about  three  or  less;  or  taking  the 

^r 

values  of  the  circuit  constants  from  tests  of  the  cable,  we  get 
values  of  the  magnitude,  per  mile  of  single  conductor,  r  =  0.41 
ohm;  L  =  0.4  X  10~3  henry;  g  =  10~6  mho,  corresponding  to  a 
power  factor  of  the  cable-charging  current,  at  25  cycles,  of 
1  per  cent;  C  =  .6  X  10~6  farad. 

Herefrom  the  following  values  are  obtained :  u  =  513,  m  =  512, 
*  --  VLC  -  15.5  X  10~6,  k0  =  m  VLC  =  7.95  X  10~3,  and  the 
critical  wave  length  is  lWo  =  790  miles,  and  the  frequency  of  an 
undamped  oscillation,  corresponding  to  lWo,  is  /0  =  81.5  cycles 
per  second. 

As  seen,  in  an  underground  high-potential  cable  the  critical 
wave  length  is  very  much  shorter  than  in  the  overhead  long- 
distance transmission  line.  At  the  same  time,  however,  the 
length  of  an  underground  cable  circuit  is  very  much  shorter  than 
that  of  a  long-distance  transmission  line,  so  that  the  critical  wave 
length  still  is  very  large  compared  with  the  greatest  wave  length 
of  an  oscillation  originating  in  the  cable,  at  least  ten  times  as 
great.  Which  means  that  the  discussion  of  the  possible  phe- 
nomena in  any  overhead  line,  under  (1),  applies  also  to  the  under- 
ground high-potential  cable  circuit. 

In  the  present  example  the  longest  standing  wave  which  may 
originate  in  the  cable  has  the  wave  length 

lw  =  80  miles, 

which  gives 

k  =  0.0785 

and 

-4=  -  5070, 
VLC 


454  TRANSIENT  PHENOMENA 

or  about  ten  times  as  large  as  m,  so  that  m  can  still  be  neglected 
in  equation  (87),  and  we  have 

=  5070, 


VLC 
or  /  =  810  cycles  per  second, 

and  the  general  equations  of  the  phenomenon  in  long-distance 
transmission  lines,  (123)  to  (125),  also  apply  as  the  general  equa- 
tions of  standing  waves  in  high-potential  underground  cable 
circuits. 

(3)  Submarine  telegraph  cable. 

18.  Choosing  the  following  values:  length  of  cable,  single 
stranded-conductor,  ground  return,  =  4000  miles;  constants  per 
mile  of  conductor:  r  =  3  ohms,  L  =  10~3  henry,  g  =  10~6mho? 
and  C  =  0.1  X  10~6  farad,  we  get  u  =  1500;  m  =  1500;  <r  -  VLC 
=  10  X  10~6,  and  k0  =  m  VLC  =  15  X  10~3,  from  which  the 
critical  wave  length  is  lWo  =  418  miles,  and  the  corresponding 
frequency  fo  =  239  cycles  per  second. 

From  the  above  it  is  seen  that  in  a  submarine  cable  the  critical 
wave  length  lWo  is  relatively  short,  so  that  in  long  submarine 
cables  standing  waves  may  appear  which  are  not  oscillatory  in 
time  but  die  out  gradually,  that  is,  are  shown  by  the  equation 
of  case  B.  In  such  cables,  due  to  their  relatively  high  resist- 
ance, the  damping  effect  is  very  great;  u  =  1500,  and  standing 
waves,  therefore,  rapidly  die  out. 

In  the  investigation  of  the  submarine  cable,  the  complete 
equations  must  therefore  be  used,  and  q  cannot  always  be 
assumed  as  large  compared  with  m  and  u,  except  when  dealing 
with  local  oscillations. 

(4)  Long-distance  overhead  telephone  circuit. 

19.  Consider  a  telephone  circuit  of  1000  miles  length,  metallic 
return,  consisting  of  two  wires  No.  4  B.  and  S.  G.,  24  inches 
distant  from  each  other. 

Calculating  in  the  same  way  as  discussed  under  (1),  the  follow- 
ing constants  per  mile  of  conductor  are  obtained:  r  =  1.31  ohms, 
L  =  1.84  X  10~3  henry,  and  C  =  .0172  X  10~6  farad. 

As  conductance,  g,  we  may  assume 

(a)  g  =  0;  that  is,  very  perfect  insulation,  as  in  dry  weather. 

(6)  g  =  2.5  X  HT6;  that  is,  slightly  leaky  line. 


STANDING  WAVES 


455 


(c)  g  =  12  X  10~6;  that  is,  poor  insulation,  or  a  leaky  line. 

(d)  g  =  40  X  10~6;   that  is,   extremely   poor  insulation,    as 
during  heavy  rain. 

The  condition  may  also  be  investigated  where  the  line  is 
loaded  with  inductance  coils  spaced  so  close  together  that  in 
their  effect  we  can  consider  this  additional  inductance  as  uni- 
formly -distributed.  Let  the  total  inductance  per  unit  length 
be  increased  by  the  loading  coils  to 

L,  =  9  X  10-3  h, 

or  about  five  times  the  normal  value. 

Denoting  then  the  constants  of  the  loaded  line  by  the  index  1, 
we  have: 


Quantity 

(a) 

(&) 

(c) 

(d) 

u   = 

356 

429 

706 

1,518 

u,  = 

73 

146 

423 

1,236 

m  = 

356 

283 

6 

-    806 

•H- 

73 

0 

-277 

-1,090 

o-  =  VYc     = 

5.63X10-6 

•T^VZ^       = 

12.45X10-6 

k0  =  m  VLC  = 

2X10-3 

1.6X10-3 

33.7    X10~6 

4.56X10-' 

*oi=myz^'= 

910X10-6 

0 

3.45X10-3 

13.5    X10-3 

/«'o=  l£      = 

3,140 

3,920 

187,000 

1,380 

/-01=r      - 

6,900 

oo 

1,820 

464 

/•-£      = 

55.6 

45 

0.96 

128 

'~-£      = 

11.6 

0 

44 

173 

In  a  long-distance  telephone  line,  distributed  leakage  up  to  a 
certain  amount  increases  the  critical  wave  length  and  thus 
makes  even  the  long  wave  oscillatory.  Beyond  this  amount 
leakage  again  decreases  the  wave  length.  Distributed  induc- 
tance, as  by  loading  the  line,  increases  the  critical  wave  length 
if  the  leakage  is  small,  but  in  a  very  leaky  line  it  decreases  the 
critical  wave  length,  and  the  amount  of  leakage  up  to  which  an 
increase  of  the  critical  wave  length  occurs  is  less  in  a  loaded  line, 
that  is,  in  a  line  of  higher  inductance. 


456  TRANSIENT  PHENOMENA 

In  other  words,  a  moderate  amount  of  distributed  leakage 
improves  a  long-distance  telephone  line,  an  excessive  amount  of 
leakage  spoils  it.  An  increase  of  inductance,  by  loading  the  line, 
improves  the  line  if  the  leakage  is  small,  but  may  spoil  the  line 
if  the  leakage  is  considerable.  The  amount  of  leakage  up  to 
which  improvement  in  the  telephone  line  occurs  is  less  in  a 
loaded  than  in  an  unloaded  line ;  that  is,  a  loaded  telephone  line 
requires  a  far  better  insulation  than  an  unloaded  line. 


CHAPTER  IV. 

TRAVELING   WAVES. 

20.  As  seen  in  Chapter  III,  especially  in  electric  power  cir- 
cuits, overhead  or  underground,  the  longest  existing  standing 
wave  has  a  wave  length  which  is  so  small  compared  with  the 
critical  wave  length  —  where  the  frequency  becomes  zero  —  that 
the  effect  of  the  damping  constant  on  the  frequency  and  the 
wave  length  is  negligible.  The  same  obviously  applies  also  to 
traveling  waves,  generally  to  a  still  greater  extent,  since  the 
lengths  of  traveling  waves  are  commonly  only  a  small  part  of  the 
length  of  the  circuit.  Usually,  therefore,  in  the  discussion  of 
traveling  waves,  the  effect  of  the  damping  constants  on  the  fre- 
quency constant  q  and  the  wave  length  constant  k  can  be 
neglected,  that  is,  frequency  and  wave  length  assumed  as  inde- 
pendent of  the  energy  loss  in  the  circuit. 

Usually,  therefore,  the  equations  (74)  and  (75)  can  be  applied 
in  dealing  with  the  traveling  wave. 

In  these  equations  the  distance  traveled  by  the  wave  per 
second  is  used  as  unit  length  by  the  substitution 

/I  =  <rl, 
where  <r  =  VLC, 

as  this  brings  t  and  X  into  direct  comparison  and  eliminates  h 
and  k  from  the  equations  by  the  equation  (72). 

With  this  unit  length  the  critical  value  of  k,  k0  =  m  VTU, 
by  substituting  (69)  and  (68),  gives  q0  =  m,  and  the  condition 
of  the  applicability  of  equations  (74)  and  (75),  therefore,  is  that 
q  be  a  large  quantity  compared  with  q0  =  m. 

In  this  case  —  is  a  small  quantity,  and  thus  can  usually  be 

neglected  in  equations  (76)  and  (75),  except  when  C  and  C'  are 
very  different  in  magnitude. 

467 


458 


TRANSIENT  PHENOMENA 


This  gives,  under  the  limiting  conditions  discussed  above,  the 
general  equations  of  the  traveling  wave,  thus: 

i  =  £-ut  { £+«('-*)  [(7^  cos  q  (t  —  X)  +  (7/  sin  q  (t  —  X)] 

-  £+s«+x)  [C2  cos  q  (t  +  X)  +  C2'  sin  q  (t  +  X)] 
+  £—<«-*>  [C3  cos  q(t-  X)+  C3'  sin  q  (t  -  /I)] 

-  <•-* ('+A)  [C4  cos  g  (t  +  ;)  +  C/  sin  q  (t  +  ^)] }      (141) 


and 

— v/I-- 


[Cj  cos  g  (^  —  ^)  4-  C/  sin  q  (t  —  X)] 
[C2  cos  q  (t  +  X)  +  C27  sin  g  0  +  ^)] 
[C8  cos  g  ft  -  A)  4-  C87  sin  g  (*  -  A)] 
[(74  cos  g  (^  +  X)  +  (7/  sin  g  (t  +  A)] } , 

(142) 


or 


cos  q(t 


-  /I)] 
+  X)] 


(143) 


and 


[A2  cos  q  (t  - 
[A3  cos  g  (t  + 
[A  4  cos  g  (t  — 


A/  sin  g  (* 
A/  sin  q  (t 
A±  sin  q  (t 


where 


and 


^   -  o-Z, 

a-  =  vie. 


(144) 


(145) 


In  these  equations  (141)  to  (144)  the  sign  of  ^  may  be  reversed, 
which  merely  means  counting  the  distance  in  opposite  direction. 


TRAVELING   WAVES  459 

This  gives  the  following  equations: 


i  =  £-ut {£+s(f   A)  [Bl  cos  q  (t  —  X)  +  .B/  sin  q  (t  —  X)] 

4-  £~s(<~A)  [53  cos  g  (£  —  .A)  +  jBs7  sin  q  (t  —  X)] 

and 


e  -         •-«"-"         i  cos  «      -      +        sn  «     T 

+  e+s(t+»  [B2  cos  g  (t  +  X)  +  5/  sin  q  (t  +  X)] 
+  £-s«-*)  [J58  cos  g  (t  -  X)  +  Bz'  sin  q  (t  -  /I)]  . 
+  e-*f(<+x)  [54  cos  g  (<  +  ^)+  547  sin  q  (t  +  ^)]}, 

(147) 
or 


[A2  cos  g  (t  +  X)  +  Aa7  sin  q  (t  +  X)] 
[A3  cos  ?(«->!)+  ^3r  sin  q  (t  -  X)] 
[^I4cosg  («  +  ;)+  A/sin  q  (t  +  ;)]}  (148) 

and 


[A3  cos  ?(*  +  ;)+  A/  sin  q  (t  +  X)] 
[A3  cos  g  («  -  X)  +  A3'  sin  g  (t  -  X)] 
[A4  cos  q(t  +  X)+  AJ  sin  q(t+  X)]}. 

(149) 

In  these  equations  (141)  to  (149)  the  values  A,  B,  C,  etc.,  are 
integration  constants,  which  are  determined  by  the  terminal 
conditions  of  the  problem. 

The  terms  with  (t  -  X)  may  be  considered  as  the  main  wave, 
the  terms  with  (t  +  X)  as  the  reflected  wave,  or  inversely,  depend- 
ing on  the  direction  of  propagation  of  the  wave. 

21.  As  the  traveling  wave,  equations  (141)  to  (149),  consists 
of  a  main  wave  with  variable  (t  -  X)  and  a  reflected  wave  of  the 
same  character  but  moving  in  opposite  direction,  thus  with  the 
variable  (t  +  X),  these  waves  may  be  studied  separately,  and 
afterwards  the  effect  of  their  combination  investigated. 


460  TRANSIENT  PHENOMENA 

Thus,  considering  at  first  one  of  the  waves  only,  that  with  the 
variable  (t  -  X),  from  equations  148  and  149  we  have 


[Alcosq(t  -  X)+  A/singft  -  X)] 
+  £-•«-*>  [A3  cos  q  (t  -  X)  +  Az'  sin  q  (t  -  X)}} 
e~ut  {  (Aie+8('-x)      +  A3£-S('-A))  cos  q(t-  X) 
+  (A1's+a('-A)    +  A8'e  —  ('-A))  sin  £  (t  -  X)} 

(150) 
and 


that  is,  in  a  single  traveling  wave  current  and  voltage  are  in 
phase  with  each  other,  and  proportional  to  each  other  with  an 
effective  impedance 

(152) 


This  proportionality  between  e  and  i  and  coincidence  of  phase 
obviously  no  longer  exist  in  the  combination  of  main  waves 
and  reflected  waves,  since  in  reflection  the  current  reverses  with 
the  reversal  of  the  direction  of  propagation,  while  the  voltage 
remains  in  the  same  direction,  as  seen  by  (148)  and  (149). 

In  equation  (150)  the  time  t  appears  only  in  the  term  (t  —  X) 
except  in  the  factor  e~ut,  while  the  distance  X  appears  only  in  the 
term  (t  —  X).  Substituting  therefore 


hence 

f  .<*  V-M; 

that  is,  counting  the  time  differently  at  any  point  X,  and  counting 
it  at  every  point  of  the  circuit  from  the  same  point  in  the  phase  of 
the  wave  from  which  the  time  t  is  counted  at  the  starting  point 
of  the  wave,  X  =  0,  or,  in  other  words,  shifting  the  starting  point 
of  the  counting  of  time  with  the  distance  X,  and  substituting  in 
(150),  we  have 


TRAVELING  WAVES 


461 


e  = 


(A  j  cos  qtt  +  A  /  sin  g^) 


£   stl  (A  3  cos 

£     ''  *  (Aj  COS 

g-"1  (A 3  cos 


A  3'  sin 
A  /  sin 
A  3  sin 


(153) 


The  latter  form  of  the  equation  is  best  suited  to  represent  the 
variation  of  the  wave,  at  a  fixed  point  ^  in  space,  as  function  of 
the  local  time  tL. 

Thus  the  wave  is  the  product  of  a  term  £-uA  which  decreases 
with  increasing  distance  ^,  and  a  term 

e0  -  e~utl  {e+stl  (A1  cos  qtt  +  A/  sin  q$ 
+  e~stl  (A3  cos  qtt   +  A/  sin  qtt) } 


(154) 

which  latter  term  is  independent  of  the  distance,  but  merely  a 
function  of  the  time  tt  when  counting  the  time  at  any  point  of  the 
line  from  the  moment  of  the  passage  of  the  same  phase  of  the 
wave. 

Since  the  coefficient  in  the  exponent  of  the  distance  decrement 
£~uA  contains  only  the  circuit  constant, 


but  does  not  contain  s  and  q  or  the  other  integration  constants, 
resubstituting  from  equations  (71)  to  (68), 

x  =  ai  =  i  VLO, 

we  have 

uX  =  u  \/LC  I 


where  I  is  measured  in  any  desired  length. 


462  TRANSIENT  PHENOMENA 

Therefore  the  attenuation  constant  of  a  traveling  wave  is 

},  '(155) 

and  hence  the  distance  decrement  of  the  wave, 


depends  upon  the  circuit  constants  r]  L,  g,  C  only,  but  does  not 
depend  upon  the  wave  length,  frequency,  voltage,  or  current; 
hence,  all  traveling  waves  in  the  same  circuit  die  out  at  the  same 
rate,  regardless  of  their  frequency  and  therefore  of  their  wave 
shape,  or,  in  other  words,  a  complex  traveling  wave  retains  its 
wave  shape  when  traversing  a  circuit,  and  merely  decreases  in 
amplitude  by  the  distance  decrement  e~w\  The  wave  attenua- 
tion thus  is  a  constant  of  the  circuit. 

The  above  statement  obviously  applies  only  for  waves  of  con- 
stant velocity,  that  is,  such  waves  in  which  q  is  large  compared 
with  s,  u,  and  m,  and  therefore  does  not  strictly  apply  to  ex- 
tremely long  waves,  as  discussed  in  13. 

22.  By  changing  the  line  constants,  as  by  inserting  inductance 
L  in  such  a  manner  as  to  give  the  effect  of  uniform  distribution 
(loading  the  line),  the  attenuation  of  the  wave  can  be  reduced, 
that  is,  the  wave  caused  to  travel  a  greater  distance  /  with  the 
same  decrease  of  amplitude. 

As  function  of  the  inductance  L,  the  attenuation  constant  (155) 
is  a  minimum  for 

— °=o- 

dL 
hence, 

rO  -  gL  =  0, 
or 

(156) 


and  if  the  conductance  g  =  0  we  have  L  =  <x> ;  hence,  in  a  per- 
fectly insulated  circuit,  or  rather  a  circuit  having  no  energy  losses 
depending  on  the  voltage,  the  attenuation  decreases  with  increase 
of  the  inductance,  that  is,  by  "loading  the  line,"  and  the  more 
inductance  is  inserted  the  better  the  telephonic  transmission. 


TRAVELING  WAVES  463 

In  a  leaky  telephone  line  increase  of  inductance  decreases  the 
attenuation,  and  thus  improves  the  telephonic  transmission,  up 
to  the  value  of  inductance, 

L  =  -,  (157) 

9 

and  beyond  this  value  inductance  is  harmful  by  again  increasing 
the  attenuation. 

For  instance,  if  a  long-distance  telephone  circuit  has  the 
following  constants  per  mile:  r  =  1.31  ohms,  L  =  1.84  X  10~3 
henry,  g  =  1.0  X  1Q-"  mho,  and  C  =  0.0172  X  10~6  farad,  the 
attenuation  of  a  traveling  wave  or  impulse  is 

u0  =  0.00217; 

hence,  for  a  distance  or  length  of  line  of  1Q  =  2000  miles, 
e-nA  =  £-4.34  =  0.0129; 

that  is,  the  wave  is  reduced  to  1.29  per  cent  of  its  original  value. 
The  best  value  of  inductance,  according  to  (157),  is 

L  =  -C  =  0.0225  henry, 

i/ 

and  in  this  case  the  attenuation  constant  becomes 

u,  =  0.00114, 

and  thus 

e-«A  =  £-2.24  =  0.1055, 

or  10.55  per  cent  of  the  original  value  of  the  wave;  which  means 
that  in  this  telephone  circuit,  by  adding  an  additional  inductance 
of  22.5  -  1.84  =  20.7  mh.  per  mile,  the  intensity  of  the  arriving 
wave  is  increased  from  1.29  per  cent  to  10.55  per  cent,  or  more 
than  eight  times. 

If,  however,  in  wet  weather  the  leakage  increases  to  the  value 
g  =  5  x  10~8,  we  have  in  the  unloaded  line 

UQ  =  0.00282  and  e~^  =  0.0035, 

while  in  the  loaded  line  we  have 

u0  =  0.00341  and  e~u°l  =  0.0011, 


464  TRANSIENT  PHENOMENA 

and  while  with  the  unloaded  line  the  arriving  wave  is  still  0.35 
per  cent  of  the  outgoing  wave,  in  the  loaded  line  it  is  only  0.11 
per  cent;  that  is,  in  this  case,  loading  the  line  with  inductance 
has  badly  spoiled  telephonic  communication,  increasing  the 
decay  of  the  wave  more  than  threefold.  A  loaded  telephone  line, 
therefore,  is  much  more  sensitive  to  changes  of  leakage  g,  that  is, 
to  meteorological  conditions,  than  an  unloaded  line. 
23.  The  equation  of  the  traveling  wave  (153), 


e+*  (Al  cos  qtt  +  A/  sin  gtt) 
e~sti(A3  cos  gti  +  A3'  sin  qtt)  }  , 


e  =  e-t*  e-«*  { 

+ 

can  be  reduced  to  the  form 

e  =  e-^jj^fi-^  (fi+*i  -  e-*i)  Smqth 

+  Ef-***  (e+^  -  e-'S  cos  qtk  }  ,          (158) 
where 

k  =  ^  -  7i  =  *  ~  *  -  ?i   1 
and  L  (159) 


By  substituting  (159)  in  (158);  expanding,  and  equating  (158) 
with  (153),  we  get.  the  identities 


(160) 


Eje~8yi  cos  qy^  —  E2e~sy*  sin  qy2  =  Av 
E^'^1  sin  qy{  +  E2£~sy*  cosqy2  =  A/, 
E^+Syi  cos  qy^  —  E2e+Sy*  sin  qy2  =  -  A3, 
^iS4"^1  sin  gyt  +  #2e+SY»  cos  gy2  =  -A/, 
and  these  four  equations  determine  the  four  constants  Ev  E2, 

Any  traveling  wave  can  be  resolved  into,  and  considered  as 
consisting  of,  a  combination  of  two  waves: 
the  traveling  sine  wave. 


£-*V)  sin  qt^  (161) 

and  the  traveling  cosine  wave, 

e2  =  E2e~^  £-«'«.  (e+^  -  e-*4)  cos  ^.  (162) 


TRAVELING  WAVES  465 

Since  q  is  a  large  quantity  compared  with  u  and  s,  the  two 
component  traveling  waves,  (161)  and  (162),  differ  appreciably 
from  each  other  in  appearance  only  for  very  small  values  of  th 
that  is,  near  tti  =  0  and  tlt  =  0.  The  traveling  sine  wave  rises 
in  the  first  half  cycle  very  slightly,  while  the  traveling  cosine  wave 
rises  rapidly;  that  is,  the  tangent  of  the  angle  which  the  wave 

de 
makes  with  the  horizontal,  or—  ,  equals  0  with  the  sine  wave  and 

has  a  definite  value  with  the  cosine  wave. 

All  traveling  waves  in  an  electric  circuit  can  be  resolved  into 
constituent  elements,  traveling  sine  waves  and  traveling  cosine 
waves,  and  the  general  traveling  wave  consists  of  four  component 
waves,  a  sine  wave,  its  reflected  wave,  a  cosine  wave  and  its 
reflected  wave. 

The  elements  of  the  traveling  wave,  the  traveling  sine  wave  ev 
and  the  traveling  cosine  wave  e2  contain  four  constants:  the 
intensity  constant,  E\  the  attenuation  constant,  u,  and  u0 
respectively;  the  frequency  constant,  q,  and  the  constant,  s. 

The  wave  starts  from  zero,  builds  up  to  a  maximum,  and  then 
gradually  dies  out  to  zero  at  infinite  time. 

The  absolute  term  of  the  wave,  that  is,  the  term  which  repre- 
sents the  values  between  which  the  wave  oscillates,  is 


+stl  -  e~s^.  (163) 


The  term  e0  may  be  called  the  amplitude  of  the  wave.    It  is  a 

maximum  for  the  value  of  tl}  given  by 

< 


which  gives 

_  (u  _  8)  ,-<*-•>>  +  (U  +  S) 

hence, 


u  —  s 
and 


1          u  +  s 

2s         u  —  s 


466  TRANSIENT  PHENOMENA 

and  substituting  this  value  into  the  equation  of  the  absolute 
term  of  the  wave,  (163),  gives 


u 
9  Q  hi    4-    <?\       2s 

/W  +  SN  (165) 


V3^?  \u  -  » 

The  rate  of  building  up  of  the  wave,  or  the  steepness  of  the  wave 
front,  is  given  by 


0  n-o 

as 

G.  =  Ee~^[-  (u  -  s)  £-<M-^<  +  (u  +  s) 

=  2s#£-MA;  (166) 

that  is,  the  constant  s,  which  above  had    no   interpretation, 
represents  the  rapidity  of  the  rise  of  the  wave. 

Referring,  however,  the  rise  of  the  wave  to  the  maximum 
value  em  of  the  wave,  and  combining  (165)  with  (166),  we  have 

u+s 

G   -  em  ^±  -  (167) 


(u  -  s)  2s 

The  rapidity  of  the  rise  of  the  wave  is  a  maximum,  that  is, 
tt  a  minimum,  for  the  value  of  s,  in  equation  (164),  given  by 


ds 
which  gives 

u  +  s        2  us 


log 


u 


hence,  s  =  0,  or  the  standing  wave,  which  rises  infinitely  fast, 
that  is,  appears  instantly. 

The  smaller  therefore  s  is,  the  more  rapidly  is  the  rise  of  the 
traveling  wave,  and  therefore  s  may  be  called  the  acceleration 
constant  of  the  traveling  wave. 

24.  In  the  components  of  the  traveling  wave,  equations  (161) 
and  (162),  the  traveling  sine  wave, 

el  =  #£-«*€-«"' (£+•*'  -  e-*)  sin^z  (161), 


TRAVELING   WAVES  467 

and  the  traveling  cosine  wave,  (162), 

e2  =  Ez~u^-utl  (e+sil  -  e-*)  cos  fa 

with  the  amplitude, 

e0  =  Ee~u^-ut<  (e+st>  -  e~stl),  (168) 

we  have 

el  =  e0  sin  qlt    1 
and  (169) 

e2  =  e0  cos  qtt.  J 

If  ^  =  0,  e0  =  0;  that  is,  tt  is  the  time  counted  from  the 
beginning  of  the  wave.  » 

It  is 

tt  =  t  -  X  -  r, 

or,  if  we  change  the  zero  point  of  distance,  that  is,  count  the 
distance  X  from  that  point  of  the  line  at  which  the  wave  starts 
at  time  t  =  0,  or,  in  other  words,  count  time  t  and  distance  X 
from  the  origin  of  the  wave, 

tt  =  t  -  ;, 
and  the  traveling  wave  thus  may  be  represented  by  the  amplitude, 


the  sine  wave, 

el  =  Ee~ut  (s+stl  —  s~stl)  sin  qti  =  eQ  sin  qti\ 
the  cosine  wave, 

e2  =  Es~wt  (s+stl  —  e~stl)  cos  qti  =  e0  cos  qtt', 


(170) 


and  ti  =  t  —  X  can  be  considered  as  the  distance,  counting 
backwards  from  the  wave  front,  or  the  temporary  distance;  that 
is,  distance  counted  with  the  point  X,  which  the  wave  has  just 
reached,  as  zero  point,  and  in  opposite  direction  to  X. 

Equation  (170)  represents  the  distribution  of  the  wave  along 
the  line  at  the  moment  t. 

As  seen,  the  wave  maintains  its  shape,  but  progresses  along 
the  line,  and  at  the  same  time  dies  out,  by  the  time  decrement 


468  TRANSIENT  PHENOMENA 

Resubstituting, 

ti  =  t  -  X, 
the  equation  of  the  amplitude  of  the  wave  is 


(171) 


As  function  of  the  distance  X,  the  amplitude  of  the  traveling 
wave,  (171),  is  a  maximum  for 


dX 


which  gives 


X  =  0; 


that  is,  the  amplitude  of  the  traveling  wave  is  a  maximum  at  all 
times  at  its  origin,  and  from  there  decreases  with  the  distance. 
This  obviously  applies  only  to  the  single  wave,  but  not  to  a 
combination  of  several  waves,  as  a  complex  traveling  wave.  - 


For 


X  =  0, 


and  as  function  of  the  time  t  this  amplitude  is  a   maximum, 
according  to  equations  (163)  to  (165),  at 


and  is 


1         u  +  s 
2s        u  —  s 

2s       /u  + 


(172) 


At  any  other  point  X  of  the  circuit,  the  amplitude  therefore 
is  a  maximum,  according  to  equation  (164),  at  the  time 


and  is 


tm  - 


em  =  E 


2*-*  /^y 

Vw2  -  s2  w  -  s/ 


(173) 


TRAVELING   WAVES 


469 


25.  As  an  example  may  be  considered  a  traveling  wave  having 
the  constants  u  =  115,  s  =  45,  q  =  2620,  and  E  =  100,  hence, 


where  tt  =  t  —  L 

In  Fig.  99  is  shown  the  amplitude  eQ  as  function  of  the  dis- 
tance X,  for  the  different  values  of  time, 

t  =  2,  4,  8,  12,  16,  20,  24,  and  32  X  10~3, 


v 


16,  2^ 


Distin 


4         6         8        10        12       14        16        18        20       22        24 

Fig.  99.    Spread  of  amplitude  of  electric  traveling  wave. 

with  the  maximum  amplitude  em,  in  dotted  line,  as  envelope  of 
the  curve  of  eQ.  - 

As  seen,  the  amplitude  of  the  wave  gradually  rises,  and  at  the 
same  time  spreads  over  the  line,  reaching  the  maximum  at  the 
starting  point  A  =  0  at  the  time  t0  =  9.2  X  10~3  sec.,  and  then 
decreases  again  while  continuing  to  spread  over  the  line,  until  it 
gradually  dies  out. 

It  is  interesting  to  note  that  the  distribution  curves  of  the 
amplitude  are  nearly  straight  lines,  but  also  that  in  the  present 
instance  even  in  the  longest  power  transmission  line  the  wave  has 
reached  the  end  of  the  line,  and  reflection  occurs  before  the 
maximum  of  the  curve  is  reached.  The  unit  of  length  A  is  the 
distance  traveled  by  the  wave  per  second,  or  188,000  miles,  and 
during  the  rise  of  the  wave,  at  the  origin,  from  its  start  to  the 
maximum,  or  9.2  X  10~3  sec.,  the  wave  thus  has  traveled  1760 
miles,  and  the  reflected  wave  would  have  returned  to  the  origin 
before  the  maximum  of  the  wave  is  reached,  providing  the  cir- 
cuit is  shorter  than  880  miles. 


470 


TRANSIENT  PHENOMENA 

2    4    6    8    10    12   14   16    18   20   22   21 


34   36   38   40   42   44   46   48 

Fig.  100.    Passage  of  traveling  wave  at  a  given  point  of  a  transmission  line 


U.5 


°*(qti 


1.0 


+  45°) 


2  5.2621 


2.0 


2.5 


\ 


\ 


SOX  10'a  Sec. 


Fig.  101.     Beginning  of  electric  traveling  waves. 


TRAVELING  WAVES  471 

With  s  =  1  it  would  be  t0  =  8.7  X  10"3  sec.,  or  nearly  the 
same,  and  with  s  =  0.01  it  would  be  t0  =  3.75  X  10~3  sec.,  or, 
in  other  words,  the  rapidity  of  the  rise  of  the  wave  increases  very 
little  with  a  very  great  decrease  of  s. 

Fig.  100  shows  the  passage  of  the  traveling  wave,  el  =  e0  sin  qtb 
across  a  point  X  of  the  line,  with  the  local  time  tt  as  abscissas 
and  the  instantaneous  values  of  el  as  ordinates.  The  values  are 
given  for  ^  =  0,  where  tt  =  t]  for  any  other  point  of  the  line  X 
the  wave  shape  is  the  same,  but  all  the  ordinates  reduced  by  the 
factor  £~115*  in  the  proportion  as  shown  in  the  dotted  curve  in 
Fig.  99. 

Fig.  101  shows  the  beginning  of  the  passage  of  the  traveling 
wave  across  a  point  X  =  0  of  the  line,  that  is,  the  starting  of  a 
wave,  or  its  first  one  and  one-half  cycles,  for  the  trigonometric 
functions  differing  successively  by  45  degrees,  that  is, 

e    =  e  sin   t 


cos  qtt  =  e0  sin  (  qtt  +  ^ 


=  6  cos        +      =  e   sin 


The  first  curve  of  Fig.  101  therefore  is  the  beginning  of  Fig.  100. 

In  waves  traveling  over  a  water  surface  shapes  like  Fig.  101 
can  be  observed. 

For  the  purpose  of  illustration,  however,  in  Figs.  100  and  101 
the  oscillations  are  shown  far  longer  than  they  usually  occur; 
the  value  q  =  2620  corresponds  to  a  frequency  /  =  418  cycles, 
while  traveling  waves  of  frequencies  of  100  to  10,000  times  as 
high  are  more  common. 

Fig.  102  shows  the  beginning  of  a  wave  having  ten  times  the 
attenuation  of  that  of  Fig.  101,  that  is,  a  wave  of  such  rapid 
decay  that  only  a  few  half  waves  are  appreciable,  for  values  of 
the  phase  differing  by  30  degrees. 

26.  A  specially  interesting  traveling  wave  is  the  wave  in 
which 

s  =  u,  (174) 


472 


TRANSIENT  PHENOMENA 


since  in  this  wave  the  time  decrement  of  the  first  main  wave  and 
its  reflected  wave  vanishes, 

e-«--*  =  1;  (175) 

that  is,  the  first  main  wave  and  its  reflected  wave  are  not  tran- 
sient but  permanent  or  alternating  waves,  and  the  equations  of 


12  16 


\ 


oanfqti 


^edooaqtj 


+  60 


h 


±± 


+  60 


20 


Time  t 


=,1150 


\ 


LiM 


X 


Fig.  102.    Passage  of  a  traveling  wave  at  a  given  point  of  a  line. 

the  first  main  wave  give  the  equations  of  the  alternating-current 
circuit  with  distributed  r,  L,  g,  C,  which  thus  appear  as  a  special 
case  of  a  traveling  wave. 

Since  in  this  case  the  frequency,  and  therewith  the  value  of  q, 
are  low  and  comparable  with  u  and  s,  the  approximations  made 


TRAVELING  WAVES  473 

in  the  previous  discussion  of  the  traveling  wave  are  not  per- 
missible, but  the  general  equations  (50)  and  (51)  have  to  be  used. 
Substituting  therefore  in  (50)  and  (51), 

s  =  u, 
gives 

i  =  [£~hl  {Cl  cos  (qt  -  kl)  +  C/  sin  (qt  -  kl) } 
-  e+hl  \C2  cos  (qt  +  kl)  +  C/  sin  (qt  +  kl)}] 
-  r2ut  [£~hl  {C4  cos  (qt  +  kl)  +  C4'  sin  (qt  +  kl) } 


-    +hl 
and 


-  e       {C3  cos  (qt  -  kl)  +  <73'  sin  (qt  -  kl)}]         (176) 


-  (c/C,  +  c/7/)  sin  (qt  -  kl)  } 
l  {  (c/CY-  ctC2)   cos  (qt  +  kl) 


hl  {  (c2'C4  -  C2C4)   cos  (qt  +  kl) 


+  £          c,   -  c23  cos  $  -   ; 

-(C/C3  +  c2(73Osin(^-^)}].        (177) 

In  these  equations  of  current  i  and  e.m.f.  e  the  first  term 
represents  the  usual  equations  of  the  distribution  of  alternating 
current  and  voltage  in  a  long-distance  transmission  line,  and  can 
by  the  substitution  of  complex  quantities  be  reduced  to  a  form 
given  in  Section  III. 

The  second  term  is  a  transient  term  of  the  same  frequency; 
that  is,  in  a  long-distance  transmission  line  or  other  circuit  of 
distributed  r,  L,  gt  C,  when  carrying  alternating  current  under  an 
alternating  impressed  e.m.f.,  at  a  change  of  circuit  conditions,  a 
transient  term  of  fundamental  frequency  may  appear  which  has 
the  time  decrement,  that  is,  dies  out  at  the  rate 


In  this  decrement  the  factor 


474  TRANSIENT  PHENOMENA 

is  the  usual  decrement  of  a  circuit  of  resistance  r  and  inductance 
Lj  while  the  other  factor, 


may  be  attributed  to  the  conductance  and  capacity  of  the  circuit, 
and  the  total  decrement  is  the  product, 


A  further  discussion  of  the  equations  (176)  and  (177)  and  the 
meaning  of  their  transient  term  requires  the  consideration  of 
the  terminal  conditions  of  the  circuit. 

27.  The  alternating  components  of  (176)  and  (177), 

io  =  s-ja{C1  cos  (qt  -  kl)  +  C/  sin  (qt  -  kl) } 

-  e+hl{C2  cos  (qt  +  kl)  +  C/  sin  (qt  +  kl) }        (178) 
and 

/>     .     *      fit     \   (  f+   '§*    *     />   (i     \   r*r\cy    ( nl 1/*7\  /*/»   '/^     ^l_  />   /•    f\    01  T~I    /'/v/  7^7\   (. 

t/fl  —    c  i   I  O-    vy  .|       '         O^vy  -.  /   l^\_/o    \tyt/          IV v J          V^i    ^  i  ~1~  ^•|^-/  -i    /    olll    I  Ul>  —  fi/L )  I 

(179) 

are  reduced  to  their  usual  form-  in  complex  quantities  by  resolv- 
ing the  trigonometric  function  into  functions  of  single  angles, 
qt  and  kl,  then  dropping  cos  qt,  and  replacing  sin  qt  by  the  imagi- 
nary unit  j.  This  gives 

i0  =  s~hl  { (Cl  cos  kl  —  Of  sin  kl)  cos  qt' 

+  (C/  cos  kl  +  Cj  sin  A;Z)  sin  ^} 

—  e+hl  { ((72  cos  kl  +  C27  sin  A;Q  cos  qt 

+  (C2  cos  kl  —  C2    sin  A^Z)  sin  qt } ; 
hence,  in  complex  expression, 
/  =  e~hl  { (C1  +  yc/)  cos  &Z  —  (C/  —  yCJ  sin  A;Z} 

-  e+w  { (Ca  +  yCa7)  cos  kl  +  (C/  -  yCa)  sin  &£},  (180) 

and  in  the  same  manner, 

+  [c/  (c,  +  yc/)  +  Cl  (C/-  ycj]  sin  w} 

-  [c/  (Ca  +  yc/)  +  c\  (C/  -  j(722)]  sin  JW } .  (181) 


TRAVELING  WAVES 


475 


However,  from  equation  (52), 

qk  +  h  (m  +  s)  T       _  j       ,       k  (m  +  s)  —  qh 


since 


q  =  2 


and 


lr 


1/r 


and 


we  have 


s  4-  m  =  — 


and 


c/  = 


xk  +  rh 
h2  +  k2        =  h2  +  k2 

rk  —  2  TT/L/Z,      rk  —  xh 
h2  +  k2        =  h2  +  k2  ' 

where  x  =  2  TT/L  =  reactance  per  unit  length. 
From  equation  (54), 

R2  =  V(s2  +  q2  -  m2)2  +  4<fm2; 
hence,  substituting  (182)  and  (184)  and  also 

b  =  2  TtfC, 
we  have 


1 


zy 
LC' 


where 
and 


z  =  Vr*  +  x2  --=  impedance  per  unit  length 
y  =  Vg2  +  b2  =  admittance  per  unit  length. 


(182) 


(183) 


(184) 


(185) 


(186) 


476  TRANSIENT  PHENOMENA 

From  the  above  it  follows  that 


and 


h  =  VLC  Vi    {R*  +  s2  -  q2  •-  m2 
=  v'i  («y  +  rg  -  xb) 


k  =      \(zy  -  rg  +  xb). 
If  we  now  substitute 


(187) 


and 

or 

and 

where 

and 


A'  =  -  *y Y,  \ 


/  -  jtf,  =  +  JB,  VY, 

Z  =  r  -  jx 


(188) 


(189) 


(190) 


Y  =g-  y&; 
in  (180)  and  (181)  we  have 

7  =  vT  [B^  (coskl-jsmkl)+B2e-hl  (coskl  +  j  sinkl)}  (191) 
and 

-  B2e~hl  (cos  &Z  +  y  sin  jfcZ) } ,    (192) 
and  substituting  (183)  gives 

h  (r-jx)+k(x  +  jr)  _  (r-jx)  (h+jk)      r-jx 
i  '  j  i  ~  Ti    i    Ti  — TT> — ; — 7~9 =  T T-      v^y^y 

*7  *     ,  1        jv»^  A  *     _JI        A»*  Aj   /i  /^ 

IV         \       /v  l(j       ~\~    A/  ft — Jfi/ 

However, 


T-  k2)  -  2jhk 

=  V(rg  -  xb)  -  jV(zy)2  -  (rg  -  xb}2 
=  V  (rg  -  xb)  -  j  V  (r2  +  x2)  (g2  +  b2)  -  (rg  - 


(rg  -  xb)  -  j  Vr^b2  +  x2g2  +  2  rgxb 
V(rg  -  xb)  -  j  (rb  +  xy) 


=  V(r  -  jx)    (jgf  -  y&); 
or  A  -  y/b  =  VZF 


(194) 


TRAVELING   WAVES  477 

substituted  in  (193)  gives 


and  (195)  substituted  in  (192)  gives 

E=  VZ  {B^+hl  (coskl-jsmkl)  -B2e~hl  (cos  JW-f  /sin  kl)}, 

(196) 

where  Bl  and  B2  are  the  complex  imaginary  integration  constants. 
Writing 

h  =  a  and  k  =  f), 
B,  =  Dl  and  B2  =   -D2 

the  equations  (191)  and  (196)  become  identical  with  the  equa- 
tions of  the  long-distance  transmission  line  derived  in  Section  III, 
equations  (22)  of  paragraph  8. 

It  is  interesting  to  note  that  here  the  general  equations  of 
alternating-current  long-distance  transmission  appear  as  a  special 
case  of  the  equations  of  the  traveling  wave,  and  indeed  can  be 
considered  as  a  section  of  a  traveling  wave,  in  which  the  accelera- 
tion constant  s  equals  the  exponential  decrement  u. 


CHAPTER  V. 
FREE  OSCILLATIONS. 

28.  The  general  equations  of  the  electric  circuit,  (50)  and  (51), 
contain  eight  terms:  four  waves:  two  main  waves  and  their 
reflected  waves,  and  each  wave  consists  of  a  sine  term  and  a 
cosine  term. 

The  equations  contain  five  constants,  namely:  the  frequency 
constant,  g;  the  wave  length  constant,  &;  the  time  attenuation 
constant,  u\  the  distance  attenuation  constant,  h,  and  the  time 
acceleration  constant,  s ;  among  these,  the  time  attenuation,  uy  is 
a  constant  of  the  circuit,  independent  of  the  character  of  the  wave. 

By  the  value  of  the  acceleration  constant,  s,  waves  may  be  sub- 
divided into  three  classes,  namely:  s  =  0,  standing  waves,  as 
discussed  in  Chapter  III;  u  >  s  >  0,  traveling  waves,  as  dis- 
cussed in  Chapter  IV;  s  =  u,  •  alternating-current  and  e.m.f. 
waves,  as  discussed  in  Section  III. 

The  general  equations  contain  eight  integration  constants  C 
and  C',  which  have  to  be  determined  by  the  terminal  condi- 
tions of  the  problem. 

Upon  the  values  of  these  integration  constants  C  and  C' 
largely  depends  the  difference  between  the  phenomena  occurring 
in  electric  circuits,  as  those  due  to  direct  currents  or  pulsating 
currents,  alternating  currents,  oscillating  currents,  inductive  dis- 
charges, etc.,  and  the  study  of  the  terminal  conditions  thus  is  of 
the  foremost  importance. 

29.  By  free  oscillations   are  understood  the  transient  phe- 
nomena occurring  in  an  electric  circuit  or  part  of  the  circuit  to 
which  neither  electric  energy  is  supplied  by  some  outside  source 
nor  from  which  electric  energy  is  abstracted. 

Free  oscillations  thus  are  the  transient  phenomena  resulting 
from  the  dissipation  of  the  energy  stored  in  the  electric  field  of 
the  circuit,  or  inversely,  the  accumulation  of  the  energy  of  the 
electric  field;  and  their  appearance  therefore  presupposes  the 
possibility  of  energy  storage  in  more  than  one  form  so  as  to  allow 

478 


FREE  OSCILLATIONS  479 

an  interchange  or  surge  of  energy  between  its  different  forms, 
electromagnetic  and  electrostatic  energy.  Free  oscillations  occur 
only  in  circuits  containing  both  capacity  C  and  inductance,  L. 

The  absence  of  energy  supply  or  abstraction  defines  the  free 
oscillations  by  the  condition  that  the  power  p  =  ei  at  the  two 
ends  of  the  circuit  or  section  of  the  circuit  must  be  zero  at  all 
times,  or  the  circuit  must  be  closed  upon  itself. 

The  latter  condition,  of  a  circuit  closed  upon  itself,  leads  to  a 
full-wave  oscillation,  that  is,  an  oscillation  in  which  the  length  of 
the  circuit  is  a  complete  wave  or  a  multiple  thereof.  With  a  cir- 
cuit of  uniform  constants  as  discussed  here  such  a  full-wave 
oscillation  is  hardly  of  any  industrial  importance.  While  the 
most  important  and  serious  case  of  an  oscillation  is  that  of  a 
closed  circuit,  such  a  closed  circuit  never  consists  of  a  uniform 
conductor,  but  comprises  sections  of  different  constants  •  generat- 
ing system,  transmission  line  and  load,  thus  is  a  complex  circuit 
comprising  transition  points  between  the  sections,  at  which  par- 
tial reflection  occurs. 

The  full-wave  oscillation  thus  is  that  of  a  complex  circuit, 
which  will  be  discussed  in  the  following  chapters. 

Considering  then  the  free  oscillations  of  a  circuit  having  two 
ends  at  which  the  power  is  zero,  and  representing  the  two  ends 
of  the  electric  circuit  by  I  =  0  and  I  —  10,  that  is,  counting  the 
distance  from  one  end  of  the  circuit,  the  conditions  of  a  free 
oscillation  are 

I  =  0,  p  =  0. 

I  =  *0,  P  =  0. 

Since  p  =  ei,  this  means  that  at  I  =  0  and  I  =  10  either  e  or  i 
must  be  zero,  which  gives  four  sets  of  terminal  conditions: 


(1)  e  =  0  at  I  =  0;  i  =  0  at  I  =  Z0. 

(2)  i  =  0  at  I  =  0;  e  =  0  at  I  =  Z0. 

(3)  e  =  0  at  I  =  0;  e  =  0  at  I  =  Z0. 

(4)  i  =  0  at  I  =  0;  i  =  0  at  I  =  L. 


(197) 


Case  (2)  represents  the  same  conditions  as  (1),  merely  with  the 
distance  I  counting  from  the  other  end  of  the  circuit  —  a  line 
open  at  one  end  and  grounded  at  the  other  end.  Case  (3)  repre- 


480  TRANSIENT  PHENOMENA 

sents  a  circuit  open  at  both  ends,  and  case  (4)  a  circuit  grounded 
at  both  ends. 

30.  In  either  of  the  different  cases,  at  the  end  of  the  circuit 
I  =  0,  either  e  =  0,  or  i  =  0. 

Substituting  I  =  0  into  the  equations  (50)  and  (51)  gives 

eo  =  fi-<«- >'{[C/  (C/  +  C2')  -  c,  (C,  +  C2)]  cos  qt 


r  '  (c1  '  -i-  r  f\     r  (C1  -\-  r  v 

i     •  (L°2     V°3      '      U4  /          °2   V°3      '      U4A 

-  [ea'  (C8  +  C4)  +  c2  (C,7  +  C/)]  sin  qt}  (198) 

and 

i0  =  s~(u~s)<  { (Cj—  C2)  cos  qt  +  (C/  —  C/)  sin  qt} 

+  e~(u+sn{(C3  -C4)cosqt+(C3'  -C4')smqt}.    (199) 

If  neither  g  nor  s  equals  zero, 
for  e0  =  0, 

c/  (C/  +  <72')  -  ct  (Ct  +  Ca)  =  0       I 
and  c/  (C,  +  C2)  +  ct  (C/  +  Ca7)  =0;      J 

hence, 

\'  \  _          '/  i  (200) 

and  for  i0  =  0, 

n      -  r  r      -  r       1 

tf»     '•   V.  j  ^4:~^3J 

c'.c'         c/-c"  J          (201) 

Substituting  in  (50)  and  (51), 
i  =  e~(u~s^  {Cj  [e~hl  cos  (g£  —  kl)  ±  s+hl  cos  (g^  +  kl)] 

+  C/  [s~w  sin  (g^  -  kl)  ±  s+hl  sin  (qt  +  kl)]} 
+  €-(«+•)«  JC3  [£+w  cos  (gi  -  kl)  ±  e~hl  cos  (g^  +  kl)] 

+  C/  [£+w  sin  (qt  -  kl)  ±  e~hl  sin  (qt  +  kl)]}      (202) 


FREE  OSCILLATIONS  481 

and 

e  =  f-<«-><  {  (c/C/-  C/7J  [e~u  cos  (?*-&/) 

T  £+wcos  (qt  +  kl)] 

-  (c/C4  +  c.C/)  [e-w  sin  (#  -  &) 

=F  £+wsin(^  +  kl)]\ 
Ca/(7a/_  C2c3)  [€+«  cos  (#-&) 
=F  e~wcos  (qt  +  kl)] 
2'03  +  c203')[e+hlsm(qt-kl) 


~hl 


sm  (qt  +  kl)]},     (203) 


where  the  upper  sign  refers  to  e  =  0,  the  lower  sign  to  i  =  0  for 
I  =  0. 

31.  In  a  free  oscillation,  either  e  or  i  must  be  zero  at  the  other 
end  of  the  oscillating  circuit,  or  at  I  =  10. 

Substituting,  therefore,  /  =  Z0  in  equations  (202)  and  (203), 
and  resolving  and  arranging  the  terms  by  functions  of  t}  the 
respective  coefficients  of 


must  equal  zero,  either  in  equation  (202),  if  i  =  0  at  I  =  10,  or  in 
equation  (203),  if  6  =  0  at  I  =  Z0,  provided  that,  as  assumed 
above,  neither  s  nor  q  vanishes. 
This  gives,  for  i  =  0  at  /  =  Z0,  from  equation  (202), 


C.  (e~M°  ±  e+hl°)  cos  kL  -  C/(e~M°  =F  e+   °)  sin  kL  =  0,1 

[  (204) 
C,  (e~Mo  q=  £+w°)  sin  kl0  +  (7/(e-^  ±  e~hl°)  cos  /cZ0  =  0,  J 

and  analogously  for  C3  and  C3'. 

In  equations  (204),  either  Cv  C/,  (73,  Cg'  vanish,  and  then  the 
whole  oscillation  vanishes,  or,  by  eliminating  Cv  and  C/  from 
equations  (204),  we  get 


C0g2  ^  +  (£-o  T  €+o)a  sin2  ^  =  0. 
hence, 


osW0  =  0 
and  - 


482 


TRANSIENT  PHENOMENA 


hence,  for  the  upper  sign,  or  if  e  =  0  for  I  =  0, 

h  =  0  and  cos  klQ  =  0, 
thus: 


(206) 


and  for  the  lower  sign,  or  if  i  =  0  for  I  =  0, 

1  (207) 


«L  =  mi. 


1 


In  the  same  manner  it  follows,  for  e  =  0  at  I  =  1Q,  from  equa- 
tion (203),  if  e  =  0  for  Z  =  0, 


thus: 

and  if  i  =  0  for  I  =  0, 

thus: 


h  =  0,     sin  kl0  =  0 
kl  —  nn. 


(208) 


0    and    cos  kl0  =  0, 

(2  n  +  1)  TT 


(209) 


From  equations  (206)  to  (209)  it  thus  follows  that  h  =  0,  that 
is,  the  free  oscillation  of  a  uniform  circuit  is  a  standing  wave. 
Also 


(2  n 


(210) 


if  e  =  0  at  one,  i  =  0  at  the  other  end  of  the  circuit,  and 

kl,  -  nn  (211) 

if  either  e  =  0  at  both  ends  of  the  circuit  or  i  =  0  at  both  ends  of 
the  circuit. 
32.   From  (210)  it  follows  that 


or  an  odd  multiple  thereof;  that  is,  the  longest  wave  which  can 
exist  in  the  circuit  is  that  which  makes  the  circuit  a  quarter- 


FREE  OSCILLATIONS  483 

wave  length.  Besides  this  fundamental  wave,  all  its  odd  multi- 
ples can  exist.  Such  an  oscillation  may  be  called  a  quarter-wave 
oscillation. 

The  oscillation  of  a  circuit  which  is  open  at  one  end,  grounded 
at  the  other  end,  is  a  quarter-wave  oscillation,  which  can  contain 
only  the  odd  harmonics  of  the  fundamental  wave  of  oscillation. 

From  (211)  it  follows  that 


or  a  multiple  thereof;  that  is,  the  longest  wave  which  can  exist 
in  such  a  circuit  is  that  wave  which  makes  the  circuit  a  half- 
wave  length.  Besides  this  fundamental  wave,  all  its  multiples, 
odd  as  well  as  even,  can  exist.  Such  an  oscillation  may  be  called 
a  half-wave  oscillation. 

The  oscillation  of  a  circuit  which  is  open  at  both  ends,  or 
grounded  at  both  ends,  is  a  half-wave  oscillation,  and  a  half-wave 
oscillation  can  also  contain  the  even  harmonics  of  the  funda- 
mental wave  of  oscillation,  and  therefore  also  a  constant  term 
forn  =  Oin  (211). 

It  is  interesting  to  note  that  in  the  half-wave  oscillation  of  a 
circuit  we  have  a  case  of  a  circuit  in  which  higher  even  harmonics 
exist,  and  the  e.m.f.  and  current  wave,  therefore,  are  not  sym- 
metrical. 

From  h  =  0  follows,  by  equation  (56), 

s=0,    if    k2  >  LCm\  ] 

and  (212) 

9  =  0,    if    k2<  LCm\  J 

The  smallest  value  of  k  which  can  exist  from  equation  (210)  is 


and,  as  discussed  in  paragraph  (15),  this  value  in  high-potential 
high-power  circuits  usually  is  very  much  larger  than  LCm2,  so 
that  the  case  q  =  0  is  realized  only  in  extremely  long  circuits, 
as  long-distance  telephone  or  submarine  cable,  but  not  in  trans- 
mission lines,  and  the  first  case,  s  =  0,  therefore,  is  of  most 
importance. 


484 


TRANSIENT  PHENOMENA 


Substituting,  therefore,  h  =  0  and  s  =  0  into  the  equation 
(52)  gives 

*-y-h-. 


and 


m 


(213) 


and  substituting  into  equations  (202)  and  (203)  of  the  free  oscilla- 
tion gives 

i  =  e~ut{A1[cos  (qt  —  kl)  ±  cos  (qt  +  kl)] 

+  A2  [sin  (qt  -  kl)  ±  sin  (qt  +  kl)]}  (214) 


and 


e  =  -£-*  {(mA2-  qAj)  [cos  (qt-kl)  =F  cos  (qt  +  kl)] 


^  qA2)  [sin  ($-fcQ  =F  sin  (qt  + 


.  (215) 


where:  A,  =  C,  +  C,  and  A  =  C/  +  <73'.  ' 

Since  k  and  therefore  #  are  large  quantities,  m  can  be  neglected 
compared  with  q,  and 

k  = 
hence 


and  the  equation  (215)  assumes,  with  sufficient  approximation, 
the  form 


t  {A,  [cos  (^  -  kl)  =F  cos  (g^  +  kl)] 
-A2  [sin  (g/  -  kl)  =F  sin  (^+  kl)]},        (216) 


where  the  upper  sign  in  (214)  and  (216)  corresponds  to  e  =  0  at 
I  =  0,  the  lower  sign  to  i  =  0  at  I  =  0,  as  is  obvious  from  the 
equations. 


FREE  OSCILLATIONS 


485 


Substituting 


A,  =  A  cos  Y    and  A2  =  A  sin  7- 


(217) 


into  (214)  and  (216)  gives  the  equations  of  the  free  oscillation, 
thus  : 


and 


e  =  — 


i  =  A£~W'{COS  (qt-kl-f)  =F  cos  (qt  +  kl  - 


-£~^{cos  (qt-kl-f)  T  cos  (qt  +  kl  -  7-)}. 


(218) 


With  the  upper  sign,  or  for  e  =  0  at  I  =  0,  this  gives 
i  =  2  A£~^  cos  kl  cos  (qt  —  r) 


and 


e=     -2AV/|e-" 


sin  kl  sin  (qt  —  f) . 


(219) 


With  the  lower  sign,  or  for  i  =  0  at  I  =  0,  this  gives 
i  =  2  As~w'  sin  &/  sin  ($  -  7-) 


and 


e  =  —  2  A\f  -e-  "*  cosklcos  (qt  - 


(220) 


33.  While  the  free  oscillation  of  a  circuit  is  a  standing  wave, 
the  general  standing  wave,  as  represented  by  equations  (139)  and 
(140),  with  four  integration  constants  Av  A,',  Av  A2',  is  not 
necessarily  a  free  oscillation. 

To  be  a  free  oscillation,  the  power  ei,  that  is,  either  e  or  i,  must 
be  zero  at  two  points  of  the  circuit,  the  ends  of  the  circuit  or  sec- 
tion of  circuit  which  oscillates. 

At  a  point  ^  of  the  circuit  at  which  e  =  0,  the  coefficients  of 
cos  qt  and  sin  qt  in  equation  (139)  must  vanish.  This  gives 

(A,  +  A2)  cos  kl,  +  (A/  -  A/)  sin  ^  =  Ol 
and       -  (A,  -  A,)  sin  kl,  +  (A/  +  A/)  cos  ^  =  0.  J 

Eliminating  sin  kl,  and  cos  kl,  from  these  two  equations  gives 
(A?  -  ,422)  +  (A*  -  A*)  =  0, 

or  (222) 

A*  +  A»-A>  +A», 


486  TRANSIENT  PHENOMENA 

as  the  condition  which  must  be  fulfilled  between  the  integration 
constants. 
The  value  Zx  then  follows  from  (221)  as 

4   '  4_   4 

-t" 


2 


At  a  point  Z2  of  the  circuit  at  which  i  =  0  the  coefficients  of 
cos  qt  and  sin  qt  in  equation  (140)  must  vanish.  This  gives,  in 
the  same  manner  as  above, 

(A*  -  A?)  +  (A*  -  A")  =  0, 
that  is,  the  same  conditions  as  (221),  and  gives  for  12  the  value 


tfln 


+      4  A    f     I       A    t 

A.2  Aj     -t-  /12 

From  (223)  and  (224)  it  follows  that 

I*  =  ~  lAr  J  (225) 


That  is,  the  angles  kll  and  kl2  differ  by  one  quarter-wave 
length  or  an  odd  multiple  thereof. 

Herefrom  it  then  follows  that  if  the  integration  constants  of  a 
standing  wave  fulfill  the  conditions 

At2  +  A/2  =  A*  +  A*  =  B\  (226) 

the  circuit  of  this  wave  contains  points  lv  distant  from  each  other 
by  a  half-wave  length,  at  which  e  =  0,  and  points  Z2,  distant  from 
each  other  by  a  half-wave  length,  at  which  i  =  0,  and  the  points 
12  are  intermediate  between  the  points  lv  that  is,  distant  there- 
from by  one  quarter-wave  length.  Any  section  of  the  circuit, 
from  a  point  ^  or  12  to  any  other  point  Zt  or  12,  then  is  a  freely 
oscillating  circuit. 

In  the  free  oscillation  of  the  circuit  the  circuit  is  bounded  by 
one  point  Zt  and  one  point  12;  that  is,  the  e.m.f.  is  zero  at  one  end 
and  the  current  zero  at  the  other  end  of  the  circuit,  case  (1)  or  (2) 
of  equation  (197),  and  the  circuit  is  then  a  quarter-  wave  or  an 
odd  multiple  thereof,  or  the  circuit  is  bounded  by  two  points  Zt 
or  by  two  points  Z2,  and  then  the  voltage  is  zero  at  both  ends  of 
the  circuit  in  the  former  case,  number  (3)  in  equation  (197),  or 


FREE  OSCILLATIONS 


487 


the  current  is  zero  at  both  ends  of  the  circuit  in  the  latter  case, 
number  (4)  in  equation  (197),  and  in  either  case  the  circuit  is  one 
half-wave  or  a  multiple  thereof. 

Choosing  one  of  the  points  ^  or  12  as  starting  point  of  the  dis- 
tance, that  is,  substituting  I  —  ^  or  I  —  12  respectively,  instead 
of  /,  in  the  equations  (139)  and  (140),  with  some  transformation 
these  equations  convert  into  the  equations  (219)  or  (220).  In 
other  words,  the  equation  (226),  as  relation  between  the  integra- 
tion constants  of  a  standing  wave,  is  the  necessary  and  sufficient 
condition  that  this  standing  wave  be  a  free  oscillation. 

34.  A  single  term  of  a  free  oscillation  of  a  circuit,  with  the  dis- 
tance counted  from  one  end  of  the  circuit,  that  is,  one  point  of 
zero  power,  thus  is  represented  by  equations  (219)  or  (220), 
respectively. 

Reversing  the  sign  of  Z,  that  is,  counting  the  distance  in  the 

opposite  direction,  and  substituting  B  =  ±  2  A  y  - ,  these 
equations  assume  a  more  convenient  form,  thus: 


for 
and 

and  for 
and 


e  =  0  at  I  =  0, 
e  =  Be'"*  sin  kl  sin  (qt  -  y) 

—  s~ut  cos  kl  cos  (qt  —  y), 

j 

i  =  0  at  I  =  0, 
e  =  Be~"*  cos  kl  cos  (gtf  —  y) 


(227) 


(228) 


:~^sin  &Zsin  (qt  -  y). 
Introducing  again  the  velocity  of  propagation  as  unit  distance, 

J       —      /T-7  1 

(229) 


from  equation  (66)  and  (229)  we  get: 
kl  =  X  Vq2  +  m2 


488  TRANSIENT  PHENOMENA 

hence,  if  m  is  small  compared  with  q} 

Id  =  qX,  (230) 

and  substituting  (229)  in  (230)  gives 

k  =  <rq  =  qVLC,  (231) 

and  from  (210)  and  (211),  for  a  quarter-wave  oscillation,  we  have 

(2  n  +  1)  TT 


and 


for  a  half-wave  oscillation, 


21, 

(2n  + 


2LVW  ' 


(232) 


L>VW 


(233) 


Denoting  the  length  of  the  circuit  in  a  quarter-wave  oscillation 


by 


and  the  length  -of  the  circuit  in  a  half-wave  oscillation  by 


(234) 


(235) 


the  wave  length  of  the  fundamental  or  lowest  frequency  of 
oscillation  is 

>10  =  4  ^  =  2  Ja;  (236) 

or  the  length  of  the  fundamental  wave,  with  the  velocity  of  prop- 
agation as  distance  unit,  in  a  quarter-wave  oscillation  is 


(237) 


and  in  a  half-wave  oscillation  is 

*,  =  2  Z0  VW. 


FREE  OSCILLATIONS 


489 


Substituting  (237)  into  (232)  and  (233)  for  a  quarter-wave 
oscillation  gives 


and 


2  7T 

q  =  (2  n  +  1)  — , 


(238) 


and  for  a  half-wave  oscillation  gives 

and 
Writing  now 


27T 


(239) 


(240) 


that  is,  representing  a  complete  cycle  of  the  fundamental  fre- 
quency, or  complete  wave  in  time,  by  6  =  2  TT,  and  a  complete 
wave  in  space  by  t  =  2  TT,  from  (239)  and  (240)  we  have 


kl 


(241) 


where  n  may  be  any  integer  number  with  a  half-wave  oscillation, 
but  only  an  odd  number  with  a  quarter-wave  oscillation. 

35.  Substituting  (241)  into  (227)  and  (228)  gives  as  the 
complete  expression  of  a  free  oscillation  the  following  equation 

A.  Quarter-  wave  oscillation. 

(a)    e  =  0  at  I  =  0  (or  r  =  0) 


e  = 
and 


«£n  sin  (2n+  I)T  sin  [(2  n  +  1)  0  -  r 


(242) 


490  TRANSIENT  PHENOMENA 

(b)  i  =  0  at  Z  =  0  (or  T  =  0) 


00 


=  £~ut        Bn  cos  (2  n  +  1)  T  cos  [(2  n  +  1)  0  -  rn] 


and 


i  =  \/T£~Ut  i>#n  sin  (2  n  +  1)  r  sin  [(2  n  +  1)6-  Tn]. 
T  *-»  o 

5.   Half-wave  oscillation. 
(a)  e  =  0  at  I  =  0  (or  r  =  0) 

n  J5n  sin  TIT  sin  (nd  —  yn) 


(243) 


and 


/(7         °° 

i  =  V  Ts~ut  Sn  -^n  cos  nr  cos  (r&0  -  /-J; 
T  L  o 

(6)  t  =  0  at  I  =  0  (or  r  =  0) 


(244; 


and 


e  =  s~         »    n  cos  nr  cos 

0 


t   i/5  -v  *:;     •  ,*     ^ 

*  ==  V  7  e       2,n#r>  sm  nr  sin  (w0  -  jj, 

*  L          Y 

where 


(245) 


ko 

2^VW 


(240) 


^0  =  4  Z0  v  L(7  in  a  quarter-wave 

t  (237) 
=  2  Z0  v  LC  in  a  half-wave  oscillation,  J 

and  MV 

£~w<  =  e  ~  .-^T.  (246) 

^0  is  the  wave  length,  and  thus  —  the  frequency,  of  the  funda- 
mental wave,  with  the  velocity  of  propagation  as  distance  unit. 

It  is  interesting  to  note  that  the  time  decrement  of  the  free 
oscillation,  e~ut,  is  the  same  for  all  frequencies  and  wave  lengths, 


FREE  OSCILLATIONS  491 

and  that  the  relative  intensity  of  the  different  harmonic  compo- 
nents of  the  oscillation,  and  thereby  the  wave  shape  of  the 
oscillation,  remains  unchanged  during  the  decay  of  the  oscillation. 
This  result,  analogous  to  that  found  in  the  chapter  on  traveling 
waves,  obviously  is  based  on  the  assumption  that  the  constants 
of  the  circuit  do  not  change  with  the  frequency.  This,  however, 
is  not  perfectly  true.  At  very  high  frequencies  r  increases,  due 
to  unequal  current  distribution  in  the  conductor,  as  discussed 
in  Section  III,  L  slightly  decreases  hereby,  g  increases  by  the 
energy  losses  resulting  from  brush  discharges  and  from  electro- 
static radiation,  etc.,  so  that,  in  general,  at  very  high  frequency 

an  increase  of  y  and  ^,  and  therewith  of  u,  may  be  expected; 
Li         C 

that  is,  very  high  harmonics  would  die  out  with  greater  rapidity, 
which  would  result  in  smoothing  out  the  wave  shape  with  increas- 
ing decay,  making  it  more  nearly  approach  the  fundamental  and 
its  lower  harmonics. 

36.  The  equations  of  a  free  oscillation  of  a  circuit,  as  quarter- 
wave  or  half -wave,  (242)  to  (245),  still  contain  the  pairs  of  inte- 
gration constants  Bn  and  yn,  representing,  respectively,  the 
intensity  and  the  phase  of  the  nth  harmonic. 

These  pairs  of  integration  constants  are  determined  by  the  ter- 
minal conditions  of  time ;  that  is,  they  depend  upon  the  amount 
and  the  distribution  of  the  stored  energy  of  the  circuit  at  the 
starting  moment  of  the  oscillation,  or,  in  other  words,  on  the 
distribution  of  current  and  e.m.f.  at  t  =  0. 

The  e.m.f.,  e0,  and  the  current,  i0,  at  time  t  =  0,  can  be  ex- 
pressed as  an  infinite  series  of  trigonometric  functions  of  the 
distance  Z;  that  is,  the  distance  angle  T,  or  a  Fourier  series  of  such 
character  as  also  to  fulfill  the  terminal  conditions  in  space,  as  dis- 
cussed above,  that  is,  e  =  0,  and  i  =  0,  respectively,  at  the 
ends  of  the  circuit. 

The  voltage  and  current  distribution  in  the  circuit,  at  the 
starting  moment  of  the  oscillation,  t  =  0,  or,  6  =  0,  can  be 
represented  by  the  Fourier  series,  thus: 


cos  nr  "*"  ttn  sm 


and 


i0  =  JTn  (bn  cos  nr  +  6n'  sin  nr), 


o 


(247) 


492  TRANSIENT  PHENOMENA 

where 


an 


1    f27r 

-     /     en  cos  nrdr  =  2  avg  [e0  cos  TIT-JO    , 

71     ^0 


sin  WT  dr  =  2  avg  [e0  sin  nr]02  *, 


(248) 


and  analogously  for  b. 

The  expression  avg  [Fgj  denotes  the  average  value  of  the 
function  F  between  the  limits  at  and  a2. 

Since  these  integrals  extend  over  the  complete  wave  2  TT,  the 
wave  thus  has  to  be  extended  by  utilizing  the  terminal  conditions 
regarding  T,  but  the  wave  is  symmetrical  with  regard  to  I  =  0 
and  with  regard  to  I  =  10,  and  this  feature  in  the  case  of  a  quarter- 
wave  oscillation  excludes  the  existence  of  odd  values  of  n  in 
equations  (247)  and  (248). 

37.  Substituting  in  equations  (242)  to  (245), 

t  =  0,  '    0  =  0, 
and  then  equating  with  (247),  gives,  from  (242), 

00  oo 

eQ  —  2^n  Bn  sin  (2  n  +  1)  T  sin  fn  =  V  n  [an  cos  (2  n  +  I)T 
o  o 

f  a/  sin  (2  n  +  1)  r] 
and 

i0  =  \  j  Vn  5n  cos  (2n  +  1)  T  cos  ?-n  =  V«  [6n  cos  (2  n  +  1)  T 
^o  o 

+  &/  sin  (2  n  +  1)  r]; 
hence, 

<*n  =  0,      6n'  =  0, 

m 

Bn  sin  ?-n  =  an'  and   v  -5n  cos  j-n=  bn. 
*  iv 


FREE  OSCILLATIONS 

Equation  (242)  gives  the  constants 

an  =  0;    bn'  =  0, 


493 


(249) 


tan^n  =-~  y^; 

71 

in  the  same  manner  equation  (243)  gives  the  constants 
a>n   =  0;    bn  =  0, 


Bn 


(250) 


Equation  (244)  gives  the  same  values  as  (242),  and  (245)  the 
same  values  as  (243). 

Examples. 

38.  As  first  example  may  be  considered  the  discharge  of  a 
transmission  line :  A  circuit  of  length  Z0  is  charged  to  a  uniform 
voltage  E,  while  there  is  no  current  in  the  circuit.  This  circuit 
then  is  grounded  at  one  end,  while  the  other  end  remains 
insulated. 

Let  the  distance  be  counted  from  the  grounded  end,  and  the 
time  from  the  moment  of  grounding,  and  introducing  the  deno- 
tations (235). 

The  terminal  conditions  then  are: 

(a)  T  =  0    e  =  0, 


T   =  — 


0. 


(6)  at  6  =  0 

e  =  0  for  T  =  0;  e  =  E  for  T^  0, 

i  =  0  for  T  7^  0;  i  =  indefinite  for  r  =  0. 


494 


TRANSIENT  PHENOMENA 


The  distribution  of  e.m.f.,  ew  and  current,  i0,  in  the  circuit,  at 
the  starting  moment  6  =  0,  can  be  expressed  by  the  Fourier 
series  (247),  and  from  (248), 


On'  =  2  avg  [E  sin  (2  TT+  l)r]  = 


71   TT    L)  TC 


and 


and  from  (249), 
hence, 


and  tan  -„  = 


and  substituting  (252)  into  (242), 

4  E  sin  (2  n  +  1)  r  cos  (2  n  + 


and 


I    = 


cos(2n+ 


n        L 
From  (240)  it  follows  that 

i 

d  =  2x  gives  the  period, 
and  the  frequency, 


i  =  4 


(251) 


(252) 


(253) 


and  T  =  2  re  ^ves  the  wave  length, 


of  the  fundamental  wave,  or  oscillation  of  lowest  frequency  and 
greatest  wave  length. 


FREE  OSCILLATIONS 


495 


Choosing  the  same  line  constants  as  in  paragraph  16,  namely : 
Z0  =  120  miles;  r  =  0.41  ohm  per  mile;  L  =  1.95X10"3  henry 
per  mile;  g  =  .25  X  10~6  mho  per  mile,  and  C  =  0.0162  X  10~6 
farad  per  mile,  we  have 

u  =  113, 


and 


147  £-°-0485' 


and  the  fundamental  frequency  of  oscillation  is 

fl  =  371  cycles  per  second. 
If  now  the  e.m.f.  to  which  the  line  is  charged  is 

E  =  40,000  volts, 
substituting  these  values  in  equations  (253)  gives 

e  =  51,000  £-°-0485<J  {sin  r  cos  6  +  J  sin  3  r  cos  3  6 
sin  5  T  cos  5  6  +  ...},  in  volts 

{ cos  T  sin  0  +  j-  cos  3  T  sin  3  0 
+  |  cos  5  r  sin  5  0  +  ...},  in  amp. 

The  maximum  value  of  e  is 

e  =  E  =  40,000  volts, 

and  the  maximum  current  of  i  is 

i  =  I  =  115.5  amp. 
Since 

^\   sin  (2  n  —  1)  a  cos  (2  n  —  1)  b 

?*•       ^^— 

7T 


=  0,if  b--<a<b, 

2 


and 


_, 


_, 


(251) 


(255) 


496  TRANSIENT  PHENOMENA 

applying  (255)  to  (254)  we  have  at  any  point  T  of  the  line,  at  the 
time  6  given  by 

0  <  6  <  T:  e  =  Ez""*",       i  =  0. 

T  <  0  <  T  +  -  :  e  =  0;  i  =  h~~^. 

T  +   7T<d<T  +  —  . 

T  +  —  <  0  <T  +  2  TT;       e  =  J£e~M';       i  =  0,  etc. 

At  any  moment  of  time  6  one  part  of  the  line  has  voltage 
e  =  Ee~'u*'  and  zero  current,  and  the  other  part  of  the  line  has 
current  i  =  h~ut  and  zero  voltage,  and  the  clividing  line  between 

the  two  sections  of  the  line  is  at   T  =  6  ±  — ,  hence  moves 

z 

along  the  line  at  the  rate  T  =  6. 

39.  As  second  example  may.be  considered  the  discharge  of  a 
live  line  into  a  dead  line:  A  circuit  of  length  lv  charged  to 
a  uniform  voltage  E,  but  carrying  no  current,  is  connected  to  a 
circuit  of  the  same  constants,  but  of  length  12J  and  having  neither 
voltage  nor  current,  otherwise  both  circuits  are  insulated. 

Let  the  total  length  of  the  circuit  be  denoted  by 

and  let  the  time  be  counted  from  the  moment  where  the  circuits 
Zt  and  12  are  connected  together,  the  distance  from  the  beginning 
of  the  live  circuit  lv  whose  other  end  is  connected  to  the  dead 
circuit  ly 

Introduce  again  the  denotations  (240),  and  represent  the  total 
length  of  the  line  2  I  =  ^  +  12  by  r  =  TT,  then  write 

*. 

As  the  voltage  is  E  from  r  =  0  to  r  =  rv  and  0  from  T  =  rt  to 
T  =  TT,  the  mean  value  of  voltage,  or  the  voltage  which  will  be  left 
on  the  line  after  the  transient  phenomenon  has  passed,  is 


FREE  OSCILLATIONS  497 

and  the  terminal  conditions  of  voltage  and  current  are 

0-0 

e  =  E  -  e0  for  0  <  r  <  TI; 
e  =  -  e0  for  rx  <  T  <  TT, 
i  =  0. 

Proceeding  then  in  the  same  manner  as  in  paragraph  34,  in  the 
present  case  the  equations  (245)  and  (248)  apply,  and 

2  (   f Tl  Cn  } 

an  =  -  { J    (E  —  e0)  cos  nddd  —  J   eQ  cos  nO  dd  } 

_  2  E  sin  nrl 

hence, 

and 

2  E  (  rl          ut*,   sin  nr±  . ) 

e  =  —  }  —  +  e       Xn 1cos  TIT  cos  no  v , 

5         (256) 


^    _  %  /        ~  —  ut 


L         ^       n 

Choosing  the  same  line  constants  as  in  paragraph  35,  and 
assuming 

Zx  =  120  miles  and  12  =  80  miles, 

we  have 

I  =  100  miles  and  rt  =  0.6  TT. 

Let  E  =  40,000  volts,  ut  =  0.0404  0,  and  the  fundamental 
frequency  of  oscillation,  fv  =  445  cycles  per  second ;  then 

e  =  24,000  +  25,500  e"0'0404 '  { sin  108°  cos  T  cos  6  +  i  sin  216° ' 
cos  2  r  cos  2  0  + J  sin  348°  cos  3  T  cos  3  0  +  •  •  • }  volts 


and 

i  =  73.5  e-o-^'jsm  108°  sin  T  sin  0  +  J  sin  216°  sin  2  r 
sin  2  0+  J  sin  348°  sin  3  T  sin  3  0  +  •  •  • }  amp. 


(257) 


CHAPTER   VI. 

TRANSITION  POINTS  AND  THE  COMPLEX  CIRCUIT. 

40.  The  discussions  of  standing  waves  and  free  oscillations  in 
Chapters  III  and  V,  and  traveling  waves  in  Chapter  IV,  apply 
directly  only  to  simple  circuits,  that  is,  circuits  comprising  a  con- 
ductor of  uniformly  distributed  constants  r,  L,  g,  and  C.  Indus- 
trial electric  circuits,  however,  never  are  simple  circuits,  but  are 
always  complex  circuits  comprising  sections  of  different  con- 
stants, —  generator,  transformer,  transmission  lines,  and  load,  — 
and  a  simple  circuit  is  realized  only  by  a  section  of  a  circuit,  as 
a  transmission  line  or  a  high-potential  transformer  coil,  which  is 
cut  off  at  both  ends  from  the  rest  of  the  circuit,  either  by  open- 
circuiting,  i  =  0,  or  by  short-circuiting,  e  =  0.  Approximately, 
the  simple  circuit  is  realized  by  a  section  of  a  complex  circuit, 
connecting  to  other  sections  of 'very  different  constants,  so  that 
the  ends  of  the  circuit  can,  approximately,  be  considered  as 
reflection  points.  For  instance,  an  underground  cable  of  low  L 
and  high  (7,  when  connected  to  a  large  reactive  coil  of  high  L 
and  low  C,  may,  approximately,  at  its  ends  be  considered  as 
having  reflection  points  i  =  0.  A  high-potential  transformer 
coil  of  high  L  and  low  C,  when  connected  to  a  cable  of  low  L 
and  high  (7,  may  at  its  ends  be  considered  as  having  reflection 
points  e  =  0.  In  other  words,  in  the  first  case  the  reactive  coil 
may  be  considered  as  stopping  the  current,  in  the  latter  case  the 
cable  considered  as  short-circuiting  the  transformer.  This 
approximation,  however,  while  frequently  relied  upon  in  engi- 
neering practice,  and  often  permissible  for  the  circuit  section  in 
which  the  transient  phenomenon  originates,  is  not  permissible  in 
considering  the  effect  of  the  phenomenon  on  the  adjacent  sections 
of  the  circuit.  For  instance,  in  the  first  case  above  mentioned, 
a  transient  phenomenon  in  an  underground  cable  connected  to 
a  high  reactance,  the  current  and  e.m.f.  in  the  cable  may  approx- 
imately be  represented  by  considering  the  reactive  coil  as  a 
reflection  point,  that  is,  an  open  circuit,  since  only  a  small  current 

498 


TRANSITION  POINTS  AND   THE  COMPLEX  CIRCUIT    499 

exists  in  the  reactive  coil.  Such  a  small  current  in  the  reactive 
coil  may,  however,  give  a  very  high  and  destructive  voltage  in  the 
reactive  coil,  due  to  its  high  L,  and  thus  in  the  circuit  beyond  the 
reactive  coil.  In  the  investigation  of  the  effect  of  a  transient 
phenomenon  originating  in  one  section  of  a  complex  circuit,  as 
an  oscillating  arc  on  an  underground  cable,  on  other  sections 
of  the  circuit,  as  the  generating  station,  even  a  very  great 
change  of  circuit  constants  cannot  be  considered  as  a  reflection 
point.  Since  this  is  the  most  important  case  met  in  industrial 
practice,  as  disturbances  originating  in  one  section  of  a  complex 
circuit  usually  develop  their  destructive  effects  in  other  sections 
of  the  circuit,  the  investigation  of  the  general  problem  of  a  com- 
plex circuit  comprising  sections  of  different  constants  thus 
becomes  necessary.  This  requires  the  investigation  of  the 
changes  occurring  in  an  electric  wave,  and  its  equations,  when 
passing  over  a  transition  point  from  one  circuit  or  section  of  a 
circuit  into  another  section  of  different  constants. 

41.  The  equations  (50)  to  (57),  while  most  general,  are  less 
convenient  for  studying  the  transition  of  a  wave  from  one  circuit 
to  another  circuit  of  different  constants,  and  since  in  industrial 
high-voltage  circuits,  at  least  for  waves  originating  in  the  circuits, 
q  and  k  are  very  large  compared  with  s  and  h,  as  discussed  in 
paragraph  16,  s  and  h  may  be  neglected  compared  with  q  and  k. 
This  gives,  as  discussed  in  paragraph  9, 

h  =  o-s, 
k  =  <rq, 


G, 


r/_r/___i/ 

Cl  C2      '  V    ~ 


(258) 


where 

er  =  \/LC,  (259) 

and  substituting 

A  =  <rZ,  (260) 

that  is, 

W  =  qX'  \  (261) 

W   =  8l,     I 


500 


TRANSIENT  PHENOMENA 


gives 

i  =  e""*  (t "*'r~^  [(?!  cos  q  (X  —  t)  +  C/  sin  q  (A  —  t)] 

cos  q  (A  +  0  +  C2'  sin  ^  (^  -f-  0] 
[C8  cos  g  (J  -  0  +  C87  sin  g  (A  -  0] 

_  £   - ,.  ,  j)  [^  cos  q  y  +  t)  +cj  sin  q(l  +  t)]}     (262) 

and 

ry  /  )         •      /\       i       /^Y   /      *  /  5  ^\1 

_  72  cos  5  (^  +  0  +  ^27  ™  ?  (^  +  0] 
+  e+s  (A-<}  [Ca  cos  g  (/I  -  0  +  Cs7  sin  g  (^  -  0] 
[C4  cos  q  (A  +  0  +  C/  sin  g  (^  +  t)]}. 


Substituting  now 


C>  +  (7/2 


C22 
C32 


(263) 


(264) 


f  =  tan  «, 


TT  =  tan  r, 

U3 

C  ' 

-+•  =  tan  ^ 


(265) 


gives 


(266) 


TRANSITION  POINTS  AND   THE  COMPLEX  CIRCUIT    501 
and 


(267) 

42.  In  these  equations  (266)  and  (267)  ^  is  the  distance 
coordinate,  using  the  velocity  of  propagation  as  unit  distance, 
and  at  a  transition  point  from  one  circuit  to  another,  where  the 
circuit  constants  change,  the  velocity  of  propagation  also  changes, 
and  thus,  for  the  same  time  constants  s  and  q,  h  and  k  also  change, 
and  therewith  kl,  but  transformed  to  the  distance  variable  I,  qX 
remains  the  same;  that  is,  by  introducing  the  distance  variable  X, 
the  distance  can  be  measured  throughout  the  entire  circuit,  and 
across  transition  points,  at  which  the  circuit  constants  change, 
and  the  same  equations  (266)  and  (267)  apply  throughout  the 
entire  circuit.  In  this  case,  however,  in  any  section  of  the  circuit, 

(268) 

where  Lt  and  Ct  are  the  inductance  and  the  capacity,  respect- 
ively, of  the  section  i  of  the  circuit,  per  unit  length,  for  instance, 
per  mile,  in  a  line,  per  turn  in  a  transformer  coil,  etc. 

In  a  complex  circuit  the  time  variable  t  is  the  same  throughout 
the  entire  circuit,  or,  in  other  words,  the  frequency  of  oscillation, 
as  represented  by  q,  and  the  rate  of  decay  of  the  oscillation,  as 
represented  by  the  exponential  function  of  time,  must  be  the 
same  throughout  the  entire  circuit.  Not  so,  however,  with  the 
distance  variable  Z;  the  wave  length  of  the  oscillation  and  its  rate 
of  building  up  or  down  along  the  circuit  need  not  be  the  same, 
and  usually  are  not,  but  in  some  sections  of  the  circuit  the  wave 
length  may  be  far  shorter,  as  in  coiled  circuits  as  transformers, 
due  to  the  higher  L,  or  in  cables,  due  to  the  higher  C.  To  extend 
the  same  equations  over  the  entire  complex  circuit,  it  therefore 
becomes  necessary  to  substitute  for  the  distance  variable  /  another 
distance  variable  X  of  such  character  that  the  wave  length  has  the 
same  value  in  all  sections  of  the  complex  circuit.  As  the  wave 

length  of  the  section  i  is  —      =  >  this  is  done  by  changing  the 


unit  distance  by  the  factor  cr  =  VLf!^    The  distance  unit  of 


502  TRANSIENT  PHENOMENA 

the  new  distance  variable  ^  then  is  the  distance  traversed  by  the 
wave  in  unit  time,  hence  different  in  linear  measure  for  the 
different  sections  of  the  circuit,  but  offers  the  advantage  of 
carrying  the  distance  measurements  across  the  entire  circuit 
and  over  transition  points  by  the  same  distance  variable  A. 

This  means  that  the  length  ^  of  any  section  i  of  the  complex 
circuit  is  expressed  by  the  length  ^  =  a-^. 

The  introduction  of  the  distance  variable  A  also  has  the  advan- 
tage that  in  the  determination  of  the  constants  r,  L,  g,  C  of  the 
different  sections  of  the  circuit  different  linear  distance  measure- 
ments I  may  be  used.  For  instance,  in  the  transmission  line, 
the  constants  may  be  given  per  mile,  that  is,  the  mile  used  as 
unit  length,  while  in  the  high-potential  coil  of  a  transformer  the 
turn,  or  the  coil,  or  the  total  transformer  may  be  used  as  unit  of 
length  I,  so  that  the  actual  linear  length  of  conductor  may  be 
unknown.  For  instance,  choosing  the  total  length  of  conductor 
in  the  high-potential  transformer  as  unit  length,  then  the  length 
of  the  transformer  winding  in  the  velocity  measure  ^  is  >10  = 
\/L0C0,  where  L0  —  total  inductance,  C0  =  total  capacity  of 
transformer. 

The  introduction  of  the  distance  variable  ^  thus  permits  the 
representation  in  the  circuit  of  apparatus  as  reactive  coils,  etc., 
in  which  one  of  the  constants  is  very  small  compared  with  the 
other  and  therefore  is  usually  neglected  and  the  apparatus 
considered  as  "massed  inductance,'7  etc.,  and  allows  the  investi- 
gation of  the  effect  of  the  distributed  capacity  of  reactive  coils 
and  similar  matters,  by  representing  the  reactive  coil  as  a  finite 
(frequently  quite  long)  section  ^0  of  the  circuit. 

43.  Let  y*0,  Av  >^2,  ...  kn  be  a  number  of  transition  points  at 
which  the  circuit  constants  change  and  the  quantities  may  be 
denoted  by  index  1  in  the  section  from  ^0  to  Xv  by  index  2  in  the 
section  from  Xl  to  X2,  etc. 

At  X  =  ^  it  then  must  be  i1  =  i2,  e1  =  e2;  thus  substituting 
A  =  ^  into  equations  (266)  and  (267)  gives 


cos  q2(^+t-d2. 
(269) 


TRANSITION  POINTS  AND  THE  COMPLEX  CIRCUIT    503 


Herefrom  it  follows  that 

«,  =  <?,;  (270) 

that  is,  the  frequency  must  be  the  same  throughout  the  entire 
circuit  as  is  obvious,  and 

u2  ±  s2  =  wx  ±  st.  (271) 

Since  u2  ^  uv  only  one  of  the  two  waves  can  exist,  the  A  B, 
or  the  C  D,  and  since  these  two  wavee  differ  from  each  other 
only  by  the  sign  of  s,  by  assuming  now  that  s  may  be  either 
positive  or  negative  we  can  select  one  of  the  two  waves,  for 
instance,  the  second  wave,  but  use  A,  B,  a,  /?  as  denotations  of 
the  integration  constants. 

44.  The  equations  (266)  and  (267)  now  assume  the  form 


= 


and 


or 


and 


cos  [q  (/I  -  t)  -  a] 


cos  [q  (^  -  0  -  a] 


cos  [q  (A  —  t)  —  a] 


(272) 


-t)-  a] 


"8*  cos  [q 


(273) 


or,  using  equations  (262)  and  (263)  instead  of  (266)  and  (267), 
the  corresponding  equations  are  of  the  form 


£~ut  \£ 


and 

e  =  V- 


[A  cos  q  (X  -  t)  +  B  sin  q  (/I  -  t)] 
[C  cos  q(l  +  t)  +  D  sin  q  (I  +  0]} 


0  [A  cos?  (A  - 


+  £>  sin 


(274) 


504  TRANSIENT  PHENOMENA 

or 

—  e~s^  [C  cos  q  (A  +  0  +  D  sin  q  (X  +'01} 

and 

j~f 
e  =\/-£-(u+s»  {s+s*[Acosq()(  -  t)+Bsinq(l-t)] 


(275) 


where  s  may  be  positive  or  negative. 
From  equation  (269)  it  then  follows  that 

ul  +  sl  =  u2  +  s2  =  M3  +  s3  =  .  .  .  =  un  +  sn  =  w0,  (276) 

where  w17  w2,  w3,  etc.,  wn  are  the  time  constants  of  the  individual 

sections  of  the  complex  circuit,  ^  (  7  +  ^  )>  an<^  uo  may  ^e  callet^ 

2  \L      LI 

the  resultant  time  decrement  of  the  complex  circuit. 

45.  Equation  (269),  by  canceling  equal  terms  on  both  sides, 
then  assumes  the  form 

A1e+'>*'  cos  [q  (^-  t)  -  «J  -  5^—  '*«  cos  [g  (^,  +  0  -  &]  = 
A2^+S2^  cos  b  (A  -  0—  «21  -  ^2^"S2Al  cos  [q  (^  +  t)  -  ft], 
and,  resolved  for  cos  <#  and  sin  qt,  this  gives  the  identities 
coB  (q^  -  aj  -  5^-'^  cos  (^x  -  ft)  • 
cos  (g/l,  -  «2)  -  52£-S2;i  cos  (q^  -  ft), 
sin  (q^  -  at)  +  J?^-'1^  sin  (g^  -  /?x)  = 
sin  fe^1  -  «a)  +  52£~S2yl1  sin  (^,  -  ft).  (277) 

These  identities  resulted  by  equating  it  =  i2  from  equation 
(272).  In  the  same  manner,  by  equating  el  and  e2  from  equation 
(272)  there  result  the  two  further  identities 


[Af****  cos  (q^  -  a,)  +  BlS-^  cos  (ql,  -  ft)} 
cos  (g^  -  «2)  +  J5,e-^  cos  (q^  -  ft)  }, 


TRANSITION  POINTS  AND   THE  COMPLEX  CIRCUIT       505 


sn 


I,  -  a2)  -  Bf^  sin  fort,  -  &) }.    (278) 


Equations  (277)  and  (278)  determine  the  constants  of  any 
section  of  the  circuit,  A2,  B2,  a2,  /?2,  from  the  constants  of  the 
next  section  of  the  circuit,  Av  Bv  av  /?r 


Let 


cos 
sin 


-  a)  ==  A'; 

-  a)  =  A"; 

-  /?)  =  B'  , 


-v& 

\  a 


Then 


2c2A2'  =  (c,  +  c2)  A/  +  (c,  -  c2)fi/, 
2c.fi/  -  (cl  +  c,)Bl'  +  (c1  -cJAt', 
2c2A2"  =  (cl  +  c,)A1"  -(c,  -c,)B,', 


and  since  by  (279) : 

A'2  + 


etc., 


substituting  herein  (281), 

4  c22A22£+2S!'1'  =  (c,  +  c2)2  At2e+2"^  +  (c,-  c2) 


(279) 


(280) 


(281) 


(282) 


(283) 


506 
and 

tan 


TRANSIENT  PHENOMENA 


ct-ca  B!     2  j  sinCg^-ft) 

JL  —  i     *  /  \ 


sm 


1    i    Cl~C2  £i     -2^  COS  (g^- ft) 

*"l r       ~r~£  ^  ^          \ 


tan 


sn 


cos        - 


tan  (g^- ft). 


(284) 


In  the  same  manner,  equating,  for  ^  =  ^,  in  equations  (275) 
the  current  t\,  corresponding  to  the  section  from  >10  to  Xv  with 
the  current  iv  corresponding  to  the  section  from  ^  to  \v  and  also 
the  e.m.fs.,  e2  =  ev  gives  the  constants  in  equations  (275)  and 
(274),  of  one  section,  ^  to  Xv  expressed  by  those  of  the  next 
adjoining  section,  X0  to  Ait  as 


where 


cos  2 
sin  2 
cos  2 
sin  2 


sin  2  g^)  } 
cos  2  g/lj  } 
sin  2  g/lj  } 
cos  2  g/ij  } 


(285) 


_ 
- 


(286) 


(287) 


TRANSITION  POINTS  AND   THE  COMPLEX  CIRCUIT      507 

46.  The  general  equation  of  current  and  e.m.f.  in  a  complex 
circuit  thus  also  consists  of  two  terms,  the  main  wave  A  in 
equations  (272),  (273),  and  its  reflected  wave  B. 

The  factor  e-(»+^  =  e-^  jn  equations  (273)  and  (275)  repre- 
sents the  time  decrement,  or  the  decrease  of  the  intensity  of 
the  wave  with  the  time,  and  as  such  is  the  same  throughout  the 
entire  circuit.  In  an  isolated  section,  of  time  constant  u,  the 
time  decrement,  from  Chapters  III  and  V,  is,  however,  e~ut;  that 
is,  with  the  decrement  e~ut  the  wave  dies  out  in  the  isolated  sec- 
tion at  the  rate  at  which  its  stored  energy  is  dissipated  by  the 
power  lost  in  resistance  and  conductance.  In  a  section  of  the 
circuit  connected  to  other  sections  the  time  decrement  e~U(*  does 
not  correspond  to  the  power  dissipation  in  the  section;  that  is, 
the  wave  does  not  die  out  in  each  section  at  the  rate  as  given  by 
the  power  consumed  in  this  section,  or,  in  other  words,  power 
transfer  occurs  from  section  to  section  during  the  oscillation  of 
a  complex  circuit. 

If  s  is  negative,  u0  is  less  than  u,  and  the  wave  dies  out  in  that 
particular  section  at  a  lesser  rate  than  corresponds  to  the  power 
consumed  in  the  section,  or,  in  other  words,  in  this  section  of  the 
complex  circuit  more  power  is  consumed  by  r  and  g  than  is  sup- 
plied by  the  decrease  of  the'  stored  energy,  and  this  section, 
therefore,  must  receive  energy  from  adjoining  sections.  Inversely, 
if  s  is  positive,  u0  >  u,  the  wave  dies  out  more  rapidly  in  that 
section  than  its  stored  energy  is  consumed  by  r  and  g]  that  is, 
a  part  of  the  stored  energy  of  this  section  is  transferred  to  the 
adjoining  sections,  and  only  a  part  —  occasionally  a  very  small 
part  —  dissipated  in  the  section,  and  this  section  acts  as  a  store 
of  energy  for  supplying  the  other  sections  of  the  system. 

The  constant  s  of  the  circuit,  therefore,  may  be  called  energy 
transfer  constant,  and  positive  s  means  transfer  of  energy  from  the 
section  to  the  rest  of  the  circuit,  and  negative  s  means  reception 
of  energy  from  other  sections.  This  explains  the  vanishing  of  s 
in  a  standing  wave  of  a  uniform  circuit,  due  to  the  absence  of 
energy  transfer,  and  the  presence  of  s  in  the  equations  of  the 
traveling  wave,  due  to  the  transfer  of  energy  along  the  circuit, 
and  in  the  general  equations  of  alternating-current  circuits. 

It  immediately  follows  herefrom  that  in  a  complex  circuit 
some  of  the  s  of  the  different  sections  must  always  be  positive, 
some  negative. 


508  TRANSIENT  PHENOMENA 

In  addition  to  the  time  decrement  s~(u  +  s)i  =  £~uot  the  waves  in 
equations  (273)  and  (275)  also  contain  the  distance  decrement 
£+sl  for  ^e  mam  wave,  e~s*  for  the  reflected  wave.  Negative  s 
therefore  means  a  decrease  of  the  main  wave  for  increasing  X,  or 
in  the  direction  of  propagation,  and  a  decrease  of  the  reflected 
wave  for  decreasing  X,  that  is,  also  in  the  direction  of  propagation; 
while  positive  s  means  increase  of  main  wave  as  well  as  reflected 
wave  in  the  direction  of  propagation  along  the  circuit.  In 
other  words,  if  s  is  negative  and  the  section  consumes  more 
power  than  is  given  by  its  stored  energy,  and  therefore  receives 
power  from  the  adjoining  sections,  the  electric  wave  decreases 
in  the  direction  of  its  propagation,  or  builds  down,  showing 
the  gradual  dissipation  of  the  power  received  from  adjoining 
sections.  Inversely,  if  s  is  positive  and  the  section  thus  supplies 
power  to  adjoining  sections,  the  electric  wave  increases  in  this 
section  in  the  direction  of  its  propagation,  or  builds  up. 

In  other  words,  in  a  complex  circuit,  in  sections  of  low 
power  dissipation,  the  wave  increases  and  transfers  power 
to  sections  of  high  power  dissipation,  in  which  the  wave 
decreases. 

This  can  still  better  be  seen- from  equations  (272)  and  (274). 
Here  the  time  decrement  e~ut  represents  the  dissipation  of  stored 
energy  by  the  power  consumed  in  the  section  by  r  and  g.  The 
time  distance  decrement,  e+s^-'>  for  the  main  wave,  £~s^+<)  for 
the  reflected  wave,  represents  the  decrement  of  the  wave  for  con- 
stant U  -  t)  or  (A  +  t)  respectively;  that  is,  shows  the  change 
of  wave  intensity  during  its  propagation.  Thus  for  instance, 
following  a  wave  crest,  the  wave  decreases  for  negative  s  and 
increases  for  positive  s,  in  addition  to  the  uniform  decrease  by 
the  time  constant  e~ut'}  or,  in  other  words,  for  positive  s  the 
wave  gathers  intensity  during  its  progress,  for  negative  s  it  loses 
intensity  in  addition  to  the  loss  of  intensity  by  the  time  con- 
stant of  this  particular  section  of  the  circuit. 

47.  Introducing  the  resultant  time  decrement  UQ  of  the  com- 
plex circuit,  the  equations  of  any  section,  (273)  and  (275),  can 
also  be  expressed  by  the  resultant  time  decrement  of  the  entire 
complex  circuit,  uw  and  the  energy  transfer  constant  of  the 
individual  section;  thus 

s  =  u0  -  u,  (288) 


TRANSITION  POINTS  AND   THE  COMPLEX  CIRCUIT     509 
and 


e 


(289) 


or 

-  £~sA  [C  cos  q  (J  +  0  +  D  sin  q  (A  +  0]} 
,j  (290) 


e  =       - 


The  constants  A,  B,  C,  D  are  the  integration  constants,  and 
are  such  as  given  by  the  terminal  conditions  of  the  problem,  as 
by  the  distribution  of  current  and  e.m.f.  in  the  circuit  at  the 
starting  moment,  for  t  =  0,  or  at  one  particular  point,  as  A  =  0. 

48.  The  constants  u0  and  q  depend  upon  the  circuit  conditions. 
If  the  circuit  is  closed  upon  itself  —  as  usually  is  the  case  with  an 
electrical  transmission  or  distribution  circuit  —  and  A  is  the  total 
length  of  the  closed  circuit,  the  equations  must  give  for  A  =  A 
the  same  values  as  for  ^  =  0,  and  therefore  q  must  be  a  complete 
cycle  or  a  multiple  thereof,  2  nn;  that  is, 

?=2--  (291) 


and  the  least  value  of  q}  or  the  fundamental  frequency  of  oscilla- 
tion, is 

4.  -  v  (292) 

A 

and 

q  =  nq0.  (293) 

If  the  complex  circuit  is  open  at  both  ends,  or  grounded  at  both 
ends,  and  thus  performs  a  half-wave  oscillation,  and  At  =  total 
length  of  the  circuit, 

q,  =          and    q  =  nq0,  (294) 


510  TRANSIENT  PHENOMENA 

and  if  the  circuit  is  open  at  one  end,  grounded  at  the  other  end, 
thus  performing  a  quarter-wave  oscillation,  and  A2  =  total  length 
of  circuit,  it  is 

3  =  (2ra"1)5»'  (295) 


while,  if  the  length  of  the  complex  circuit  is  very  great  compared 
with  the  frequency  of  the  oscillation,  q0  may  have  any  value; 
that  is,  if  the  wave  length  of  the  oscillation  is  very  short  com- 
pared with  the  length  of  the  circuit,  any  wave  length,  and  there- 
fore any  frequency,  may  occur.  With  uniform  circuits,  as  trans- 
mission lines,  this  latter  case,  that  is,  the  response  of  the  line  to 
any  frequency,  can  occur  only  in  the  range  of  very  high  fre- 
quencies. Even  in  a  transmission  line  of  several  hundred  miles7 
length  the  lowest  frequency  of  free  oscillation  is  fairly  high,  and 
frequencies  which  are  so  high  compared  with  the  fundamental 
frequency  of  the.  circuit  that,  considered,  as  higher  harmonics 
thereof,  they  overlap  (as  discussed  in  the  above),  must  be 
extremely  high  —  of  the  magnitude  of  million  cycles.  In  a  com- 
plex circuit,  however,  the  fundamental  frequency  may  be  very 
much  lower,  and  below  machine  frequencies,  as  the  velocity  of 

propagation  -      -  may  be  quite  low  in  some  sections  of  the  cir- 
VLC 

curt,  as  in  the  high-potential  coils  of  large  transformers,  and  the 
presence  of  iron  increases  the  inconstancy  of  L  for  high  frequen- 
cies, so  that  in  such  a  complex  circuit,  even  at  fairly  moderate 
frequencies,  of  the  magnitude  of  10,000  cycles,  the  circuit  may 
respond  to  any  frequency. 

49.  The  constant  UQ  is  also  determined  by  the  circuit  constants. 
Upon  u0  depends  the  energy  transfer  constant  of  the  circuit  sec- 
tion, and  therewith  the  rate  of  building  up  in  a  section  of  low 
power  consumption,  or  building  down  in  a  section  of  high  power 
consumption.  In  a  closed  circuit,  however,  passing  around  the 
entire  circuit,  the  same  values  of  e  and  i  must  again  be  reached, 
and  the  rates  of  building  up  and  building  down  of  the  wave  in  the 
different  sections  must  therefore  be  such  as  to  neutralize  each 
other  when  carried  through  the  entire  circuit;  that  is,  'the  total 
building  up  through  the  entire  complex  circuit  must  be  zero. 
This  gives  an  equation  from  which  un  is  determined. 


TRANSITION  POINTS  AND   THE  COMPLEX  CIRCUIT      511 


In  a  complex  circuit  having  n  sections  of  different  constants 
and  therefore  n  transition  points,  at  the  distances 


A 


(296) 


where  An+i  =  ^i  +  A,  and  A  =  the  total  length  of  the  circuit, 
the  equations  of  i  and  e  of  any  section  i  are  given  by  equations 
(290)  containing  the  constants  A&  Bi}  Ci}  Dt-. 

The  constants  A,  B,  C,  D  of  any  section  are  determined  by 
the  constants  of  the  preceding  section  by  equations  (285)  to 
(287).  The  constants  of  the  second  section  thus  are  determined 
by  those  of  the  first  section,  the  constants  of  the  third  section 
by  those  of  the  second  section,  and  thereby,  by  substituting  for 
the  latter,  by  the  constants  of  the  first  section,  and  in  this  manner, 
by  successive  substitutions,  the  constants  of  any  section  i  can  be 
expressed  by  the  constants  of  the  first  section  as  linear  func- 
tions thereof. 

Ultimately  thereby  the  constants  of  section  (n  +  1)  are 
expressed  as  linear  functions  of  the  constants  of  the  first  section : 


An+l  -  afAl  +  af'Bt  +  a'"^  + 
Bn+l  =  VA,  +  b"B,  +  b'"Ct  + 
Cn+l  «  cfA,  +  cf'B,  +  c"'Cl  + 


Dn+1  = 


(297) 


where  a',  a",  of" ',  a"" ',  b',  b",  etc.,  are  functions  of  st  and  ^. 

The  (n  +  l)st  section,  however,  is  again  the  first  section,  and 
it  is  thereby,  by  equations  (290)  and  (296), 


Bn+l  = 


(298) 


and  substituting  (298)  into  (297)  gives  four  symmetrical  linear 
equations  in  'Alt  Bly  Clt  Dlt  from  which  these  four  constants 
can  be  eliminated,  as  n  symmetrical  linear  equations  with 


512 


TRANSIENT  PHENOMENA 


n  variables  are  dependent  equations,   containing  an  identity, 
thus: 


>t  +  (&//  _  e-<^)  B,  +  amC,  +  ft""/^  r  0; 
c^i  +  c^i  +  (cr//  -  £+'SlA)  Ct  +  c""/),  =  0; 
d'Ai  +  d"Bi  +  d'"Ci  +  (d""  -  e+s^)  D,  =  0, 
and  herefrom 


(299) 


d'"        (d""  -  £+s'A) 


=  0.  (300) 


Substituting  in  this  determinant  equation  for  st-  the  values 
from  (276) 

Si  =  u0-  Ui  (301) 


gives  an  exponential  equation  in  uw  thus : 
F  (uQyuif.^Ci)  =  0, 


(302) 


from  which  the  value  UQ,  or  the  resultant  time  decrement  of  the 
circuit,  is  determined. 

In  general,  this  equation  (302)  can  be  solved  only  by  approxi- 
mation, except  in  special  cases. 


CHAPTER  VII. 

POWER  AND  ENERGY  OF  THE  COMPLEX  CIRCUIT. 

50.  The  free  oscillation  of  a  complex  circuit  differs  from  that 
of  the  uniform  circuit  in  that  the  former  contains  exponential 
functions  of  the  distance  A  which  represent  the  shifting  or 
transfer  of  power  between  the  sections  of  the  circuit. 

Thus  the  general  expression  of  one  term  or  frequency  of  current 
and  voltage  in  a  section  of  a  complex  circuit  is  given  by  equations 
(290); 

-  £~SA  [C  cos  q  (A  +  0  +  D  sin  q  (J  +  t)]} 
and 

/7 

+SA  [A  cos  q  (A  —  t)  -f  B  sin  q  (A  —  £)] 


where  q  =  nq0,  q0  =  —  ,  A  =  total  length  of  circuit,  expressed 

in  the  distance  coordinate  A  =  o-lt  I  being  the  distance  coordinate 
of  the  circuit  section  in  any  measure,  as  miles,  turns,  etc.,  and 
r,  L,  g,  C  the  circuit  constants  per  unit  length  of  I, 

a-  =  VIC, 


u  =  -(-=•  +  —  )  =  time  constant  of  circuit  section, 
2  YL/      C  ' 

UQ=  u  +  s  =  resultant  time  decrement  of  complex  circuit, 

s   =  u0  -  u  =  energy  transfer  constant  of  circuit  section. 

613 


514  TRANSIENT  PHENOMENA 

The  instantaneous  value  of  power  at  any  point  X  of  the  circuit 
at  any  time  t  is 

p  =  ei 

[A  cos  q(X-t)  +  B  sin  q  (X  -  t)]2 
[C  cos  q  (X  +  0  +  D  sin  g  (J  +  O]2} 


+  [e+2sA(A2-£2)  cos  2q  (X-t)  -e~2s*  (C2-D2)  cos  2  g  (l  +  t)] 
+  2  [ABe+*s*sm  2  g  (X-t)  -CDe-2'*  sin  2-g  (A  +  *)]}  ;     (303) 


that  is,  the  instantaneous  value  of  power  consists  of  a  constant 
term  and  terms  of  double  frequency  in  (X  -  t)  and  (A  +  t)  or 
in  distance  A  and  time  t. 

Integrating  (303)  over  a  complete  period  in  time  gives  the 
effective  or  mean  value  of  power  at  any  point  X  as 


p  =  r*M  {fi+2«   (A2  +  J52)  -  s-2s   (C2  +  £>2)  }  ;        (304) 

.2  *  C 

that  is,  the  effective  power  at  any  point  of  the  circuit  is  the 
difference  between  the  effective  power  of  the  main  wave  and 
that  of  the  reflected  wave,  and  also,  the  instantaneous  power 
at  any  time  and  any  point  of  the  circuit  is  the  difference  between 
the  instantaneous  power  of  the  main  wave  and  that  of  the 
reflected  wave. 

The  effective  power  at  any  point  of  the  circuit  gradually 
decreases  in  any  section  with  the  resultant  time  decrement  of 
the  total  circuit,  £-2uotf  and  varies  gradually  or  exponentially 
with  the  distance  A,  the  one  wave  increasing,  the  other  decreasing, 
so  that  at  one  point  of  the  circuit  or  circuit  section  the  effective 
power  is  zero  ;  which  point  of  the  circuit  is  a  power  node,  or  point 
across  which  no  energy  flows.  It  is  given  by 

e+2sA  (A2  +  B2)  =  e~2s*  (C2  +  D2), 
or 


A2  +  B2 

C2  +  D2 


(305) 


POWER  AND  ENERGY  OF  THE  COMPLEX  CIRCUIT      515 

The  difference  of  power  between  two  points  of  the  circuit,  ^ 
and  A2,  that  is,  the  power  which  is  supplied  or  received  (depend- 
ing upon  its  sign)  by  a  section  X'  =  A2  —  Xl  of  the  circuit,  is  given 
by  equation  (304)  as 


-  (£~2S^  -  £-28^)  (C2  +  D2)}.     (306) 

If  P0  is  >  0,  this  represents  the  power  which  is  supplied  by  the 
section  X  to  the  adjoining  section  of  the  circuit;  if  P0<  0,  this  is 
the  power  received  by  the  section  from  the  rest  of  the  complex 
circuit. 

If  sX2  and  s^  are  small  quantities,  the  exponential  function 
can  be  resolved  into  an  infinite  series,  and  all  but  the  first  term 
dropped,  as  of  higher  orders,  or  negligible,  and  this  gives  the 
approximate  value 


=  ±  2  s  (;2  _  jg  =  ±  2  SA';         (307) 
hence, 

^o  -  #y%'-**  [A*  +  B2  +  O*  +  D2}  ;          (308) 

that  is,  the  power  transferred  from  a  section  of  length  A'  to  the 
rest  of  the  circuit,  or  received  by  the  section  from  the  rest  of  the 
circuit,  is  proportional  to  the  length  of  the  section,  A',  to  its  trans- 
fer constant,  s,  and  to  the  sum  of  the  power  of  main  wave  and 
reflected  wave. 

51.  The  energy  stored  by  the  inductance  L  of  a  circuit  element 
dXj  that  is,  in  the  magnetic  field  of  the  circuit,  is 

'LV 
dwl  =-^~A 

where  U  =  inductance  per  unit  length  of  circuit  expressed  by 
the  distance  coordinate  A. 

Since  L  =  the  inductance  per  unit  length  of  circuit,  of  distance 
coordinate  Z,  and  X  =  <?l, 

u     L 

Li   =  — 


VLC 


516  TRANSIENT  PHENOMENA 

hence, 

j  4  /         "2  J  5  /QAH\ 

dw^  =  ~  V  7>  ^  (ouyj 

In  general,  the  circuit  constants  r,  L,  0,  C,  per  unit  length, 
I  =  1  give,  per  unit  length,  X  =  1,  the  circuit  constants 


L.    £.    £.   5 

j 


or 


Vic '  VLC       c    VLC 

Substituting  (290)  in  equation  (309)  gives 


(310) 


+  e~2sA  [C  cos  q  (X  +  0  +  D  sin  0  (4  +  Of 
-  2  [A  cos  q  (A  -  0  +  B  sin  0  (4  -  0] 
[C  cos  q  (A  +  0  +  D  sin  0  (J  +  01} 

+  [e+2«*(Aa  -£2)  cos  2  g  (A  -0 

+  fi-^C8-  D2)  cos  2  ^  (4  +  0] 

+  2  [A£e+2sA  sin  2  0  (4  -  0  +  CDe~2sX  sin  2  g  (4  +  0] 

-  2  [(AC  -  BD)  cos  2  04  +  (AD  +  BC)  sin  2  g/l] 

-  2  [(AC  +  £D)  cos  2qt+  (AD  -  BC)  sin  2qt]}. 

(311) 

Integrating  over  a  complete  period  in  time  gives  the  effective 
energy  stored  in  the  magnetic  field  at  point  A  as 

a  w  j       j.          i*u/j  7 

7T       ==  ^         I          — TT~  Ctfr 


(A 
-  2  [(AC  -  BD)  cos  2qX+  (AD  +  BC)  sin  2  ql]},  (312) 


POWER  AND  ENERGY  OF  THE  COMPLEX  CIRCUIT     517 

and  integrating  over  one  complete  period  of  distance  A,  or  one 
complete  wave  length,  this  gives 


(313) 

The  energy  stored  by  the  inductance  L,  or  in  the  magnetic 
field  of  the  conductor,  thus  consists  of  a  constant  part, 


dl 

a  part  which  is  a  function  of  (X  —  t)  and  (X  +  t), 

(A2  -  B2)  cos  2  q  (X  -  t) 
(C2  -  D2)  cos  2  q  (X  +  0] 


n2g(/l  -  0 

n2g(yl  +  01}  ,  (315) 

a  part  which  is  a  function  of  the  distance  X  only  but  not  of  time  t, 


cos  2  ^  +  (AD  +  BC)  sin  2  ^ 

(316) 

and  a  part  which  is  a  function  of  time  £  only  but  not  of  the 
distance  ^, 

1 

2 

(317) 

and  the  total  energy  of  the  electromagnetic  field  of  circuit  element 
dX  at  time  t  is 


Aw'rr      1     /7 
"~        =    V     £~2""'{  (4(7+BI))  cos  2  9'  +  (^0-JSC)  sin  2  qt\, 


dX        d^        dl        dX         dX 

52.   The  energy  stored  in  the  electrostatic  field  of  the  conductor 
or  by  the  capacity  C  is  given  by 

CV 
dw2  =  —  dl\ 


518  TRANSIENT  PHENOMENA 

or,  substituting  (310), 


and  substituting  in  (319)  the  value  of  e  from  equation  (290) 
gives  the  same  expression  as  (311)  except  that  the  sign  of  the 
last  two  terms  is  reversed ;  that  is,  the  total  energy  of  the  electro- 
static field  of  circuit  element  dX  at  time  t  is 

dw2       dwn       dw'      dw"      dw'" 

~df  =  ~df  +  ~dT  +  ~dA~  +  ~dT'  (32°^ 

and  adding  (318)  and  (320)  gives  the  total  stored  energy  of  the 
electric  field  of  the  conductor, 

dw      dw,      dw2      cydw^         dw' 

and  integrated  over  a  complete  period  of  time  this  gives 
«  2^  = 


dw"         dw"' 
The  last  two  terms,  —  and  — ,  thus  represent  the  energy 

which  is  transferred,  or  pulsates,  between  the  electromagnetic 
and  the  electrostatic  field  of  the  circuit;  and  the  term  —  repre- 
sents the  alternating  (or  rather  oscillating)  component  of  stored 
energy. 
53.  The  energy  stored  by  the  electric  field  in  a  circuit  section 

^,  between  A,  and  A2,  is  given  by  integrating  -  -  between  A2  and  AI} 

U/A 

as 

-  (€-2«J,  _  ^-2.^  (Ca  +  £>2)  I  .  (323) 

or,  substituting  herein  the  approximation  (307), 

1  ...  II 
2 


W  =  2         2         2         2  (324) 


POWER  AND  ENERGY  OF   THE  COMPLEX  CIRCUIT     519 

Differentiating  (324)  with  respect  to  t  gives  the  power  sup- 
plied by  the  electric  field  of  the  circuit  as 


P  =  _         =  uj>         e-^  {A2  +  B*  +  C*  +  D2},   (325) 
at 

or,  more  generally, 


p  =     2.  y£-2^  {  (£+2S2  _  £-f2)  (Aa  +  £2) 

_  (e-ad,  _  e-a^)  (C8  +  D2)}.  (326) 

54.   The  power  dissipated  in  the  resistance  r'dX  =  -     r  of  a 

VLC 

conductor  element  dX  is 

dp?  =  Mdl  (327) 

2r 


hence,  substituting  herein  equation  (318)  gives  the  power  con- 
sumed by  resistance  of  the  circuit  element  dX  as 

^      2r  (flfojo      du/  _<W     dul"  ) 
d^  "  L  \  d\  ""  d/l  "~  d^   '"  ^     Y 

and  the  power  consumed  by  the  conductance  g'dX  =     ,_  rf^ 


of  a  conductor  element  dX  is 

dp"=g'<?dX  (329) 

-¥** 

hence  the  power  consumed  by  conductance  of  circuit  element  dX 
is 

^PL  =  ^1  \  ^o  ,  &;[_  ,  du/^    du/^ ) 

and  the  total  power  dissipated  in  the  circuit  element  dX  is 


520  TRANSIENT  PHENOMENA 

where,  as  before, 

h  (332) 


and  integrating  over  a  complete  period 

ffi._4J^4»4£  (333) 

dX  dX  dX 

the  power  dissipated  in  the  circuit  thus  contains  a  constant  term, 
4  u  —  -  ,  and  a  term  which  is  a  periodic  function  of  the  distance  X, 

(JLA 

4  m  —  —  -  ,  of  double  frequency. 
dX 

Averaged  over  a  half-wave  of  the  circuit,  or  a  multiple  thereof, 
the  second  term  disappears,  and 


dX  dX  ' 

or,  substituting  (314), 


,    (334) 

thus  the  power  dissipated  in  a  section  A'  =  X2  —  Xl  of  the  circuit 
is,  by  integrating  between  limits  X\  and  X21 

p"=  ^~  \T!*~***  { 

Z  S  *   G 

+   (fi  -  2^  _  £2s^  (  (72  +  ^2)  |^  (335) 

or,  approximately, 


2^  {A2  +  B*  +  (T2  +  D2}.  (336) 

55.  Writing,  therefore, 

#2  =  (A2  +  B2  +  C2  +  £>2)    /    ,  (337) 


POWER  AND  ENERGY  OF   THE  COMPLEX  CIRCUIT      521 
the  energy  stored  in  the  electric  field  of  the  circuit  section  of  length  X  is 


^;  (338) 

2 

the  power  supply  to  the  conductor  by  the  decay  of  the  electric  field  of 
the  circuit  is 

P  =  V'#2£~2w°';  (339) 


the  power  dissipated  in  the  circuit  section  X'  by  its  effective  resist- 
ance and  conductance  is 


P»  =  ul'IPe-*1*,  (340) 

and  the  power  transferred  from  the  circuit  section  A'  to  the  rest  of 
the  circuit  is 

P   =  sl'IPe-*"*',  (341) 


u 

that  is.  —  =  ratio  of  power   dissipated  in  the  section   to  that 
u0 

supplied  to  the  section  by  its  stored  energy  of  the  electric  field. 

o 

—  =  fraction  of  power  supplied  to  the  section  by  its  electric 
u0 

field,  which  is  transferred  from  the  section  to  adjoining  sections 
(or,  if  s  <  0,  received  from  them). 

o 

-  =  ratio  of  power  transferred  to  other    sections  to  power 

dissipated  in  the  section. 

u0  -f-  u  -s-  s  thus  is  the  ratio  of  the  power  supplied  to  the  sec- 
tion by  its  electric  field,  dissipated  in  the  section,  and  transferred 
from  the  section  to  adjoining  sections. 

These  relations  obviously  are  approximate  only,  and  applicable 
to  the  case  where  the  wave  length  is  short. 

56.  Equation  (306),  of  the  power  transferred  from  a  section 
to  the  adjoining  section,  can  be  arranged  in  the  form 


(A2  +  B2)  -  e-2     (C2  +  D2)] 
(A2  +  B2)  -  £-2sA'  ((T2  +  Z)2)]}  ;  (342) 

that  is,  it  consists  of  two  parts,  thus  : 

Po>  -  li/5f-**  {e+2^  (A2  +  £2)  -  e-2'*»  (C2  +  D2)},     (343) 
2  "  C 


522  TRANSIENT  PHENOMENA 

which  is  the  power  transferred  from  the  section  to  the  next  fol- 
lowing section,  and 


p*  =  +     ye-**  {e+a.,  (A2  +  ff)_e-*      (C2  +  £>2)}?   (344) 

which  is  the  power  received  from  the  preceding  section,  and  the 
difference  between  the  two  values, 

^o  -  P,'  ~  p»f>  (345) 

therefore,  is  the  excess  of  the  power  given  out  over  that  received, 
or  the  resultant  power  supplied  by  the  section  to  the  rest  of  the 
circuit. 

An  approximate  idea  of  the  value  of  the  power  transfer  con- 
stant can  now  be  derived  by  assuming  H2  as  constant  throughout 
the  entire  complex  circuit,  which  is  approximately  the  case. 

In  this  case,  as  the  total  power  transferred  between  the  sections 
must  be  zero,  thus: 


hence,  substituting  (341), 

2X-V  =  °>  (346) 

and,  since 

O fit  ,  ^      /)/ 

*»  —   ^o         »# 

w0A  =  5)1^';  (347) 

that  is,  the  resultant  circuit  decrement  multiplied  by  the  total 
length  of  the  circuit  equals  the  sum  of  the  time  constants  of  the 
sections  multiplied  with  the  respective  length  of  the  section,  or,  if 
?!,  ?2  •  •  •  ?i  =  length  of  the  circuit  section,  as  fraction  of  the 
total  circuit  length  A, 

KO  =  Dfct*t,  (348) 

Whether  this  expression  (348)  is  more  general  is  still  unknown. 
57.  As  an  example  assume  a  transmission  line  having  the 
following  constants  per  wire  :rl  =  52;LX  =  0.21  jg^  =  40  X  10~6, 
and  Cl  =  1.6  X  1Q-6. 

Further  assume  this  line  to  be  connected  to  step-up  and  step- 
down  transformers  having  the  following   constants  per  trans- 


POWER  AND  ENERGY  OF   THE  COMPLEX  CIRCUIT       523 


former  high-potential  circuit:  r2  =  5,  L2  =  3;  g2  =  0.1  X  10  6, 
and  C2  =  0.3  X  10~6;  then 

A/  =  a1  =  vT/7i  =  0.58  X  10~3,    J2'  =  *2  =  0.95  X  10~3, 
u,  =  136,  u2  -=  1. 

The  circuit  consists  of  four  sections  of  the  lengths 

V  =  0-58  X  10~3,  A2'= 0.95  X  10~3,  V  =0.58 X  10~3,  V = 0.95 X  10~3 ; 

hence  a  total  length 

A  -  3.06  X  10-3, 

and  the  resultant  circuit  decrement  is 


M»  =  ^A 


51.6  +  0.59  -  52.2; 


hence,  sl  =  -  83.8  and  s2  =    +  51.2. 

If  now  the  current  in  the  circuit  is  i0  =  100  amperes,  the  e.m.f. 
e0  =  40,000  volts,  the  total  stored  energy  is 

W  =  ;02  (L,  +  L2)  +  e*  (C,  +  C2) 
=  32,000  +  3000  =  35,000  joules, 

and  from  equation  (338)  then  follows,  for  t  =  0, 
2  -  35,000, 


=  22.8X10', 


which  gives 


u0  =  52.2, 

IP  =  22.8  X  106, 

W  =  35,000. 


Line. 

Step-up 
Transformer. 

Line. 

Step-down 
Transformer. 

Length  of  section,     X'  = 

0.58xlO~3 

0.95X10~3 

0.58X10~3 

0.95X10-3 

Time  constant,           u  = 

136 

1 

136 

1 

Transfer  constant,      s  = 

-83.8 

+  51.2 

-83.8 

+  51.2 

Energy  of  electric 
field,                        W  = 

6.650 

10.850 

6.650 

10.850kilojoules 

Power  supplied  by 

electric  field,           P= 

690 

1132 

690 

1132  kilowatts 

Power  dissipated,   P,°  = 

1800 

22 

1800 

22  kilowatts 

Power  transferred,  P0  = 

-  1110 

1110 

-  1110 

11  10  kilowatts 

524  TRANSIENT  PHENOMENA 

Thus,  of  the  total  power  produced  in  the  transformers  by  the 
decrease  of  their  electric  field,  only  22  kw.  are  dissipated  as  heat 
in  the  transformer,  and  1110  kw.  transferred  to  the  transmission 
line.  While  the  power  available  by  the  decrease  of  the  electric 
field  of  the  transmission  line  is  only  690  kw.,  the  line  dissipates 
energy  at  the  rate  of  1800  kw.,  receiving  1110  kw.  from  the 
transformers. 


CHAPTER  VIII. 

REFLECTION   AND   REFRACTION   AT   TRANSITION    POINT. 

58.  The  general  equation  of  the  current  and  voltage  in  a  sec 
tion  of  a  complex  circuit,  from  equations  (290),  is 


-  £-sA  [C  cos  q  0*  +  0  +  D  sin  q  (A  +  0]} 
e  =  C£-Uot  {e+8*  [A  cos  g  (J  -  0  +  #  sin  g  (A  -  0] 

where 

A  =  <rl  =  distance  variable  with  velocity  as  unit; 


(290) 


C' 

u0  =  u  +  s  =  resultant  time  decrement; 

1  /  f        \ 

u  =  -\j-  +  7^)  =  time  constant,  and 

2  \/v      C/ 

s  =  energy  transfer  constant  of  section. 

At  a  transition  point  ^  between  section  1  and  section  2  the 
constants  change  by 


(285) 


B2=£~s^l{a1e+8l*1Bl  +  b1e~'1*1  (Clsin2  q^l  —  Dlcos  2$is)} 

(At  cos  2  <^1  +  #1  sin  2  g^J  } 
(A  1  sin  2  gAj — 5j  cos  2  5^) } , 

where 

Oi  =  ?i_L^     and    6j  =  ^i_Z_^  .  (286) 

625 


526 


TRANSIENT  PHENOMENA 


Choosing  now  the  transition  point  as  zero  point  of  X,  so  that 
>l<  0  is  section  1,  A>0  is  section  2,  equations  (285)  assume  the 

form 


A2  = 
B2  = 
C2  = 
D2  = 


blCv 


(349) 


From  equations  (349)  and  (286)  it  follows  that 
c2  (A*  -  C22)  =  ct  (A*  -  C,2)    1 

and  (350) 

c2  (B2  -  D2)  =  c,  (B2  -  D2).  J 

If  now  a  wave  in  section  1,  A  B,  travels  towards  transition 
point  A  =  0,  at  this  point  a  part  is  reflected,  giving  rise  to  the 
reflected  wave  C  D  in  section  1,  while  a  part  is  transmitted  and 
appears  as  main  wave  A  B  in  section  2.  The  wave  C  D  in  sec- 
tion 2  thus  would  not  exist,  as  it  would  be  a  wave  coming  towards 
A  =  0  from  section  2,  so  not  a  part  of  the  wave  coming  from 
section  1.  In  other  words,  we  can  consider  the  circuit  as  com- 
prising two  waves  moving  in  opposite  direction : 

(1)  A  main  wave  AJ$V  giving  a  transmitted  wave  A  2B2  and 

reflected  wave  CJ)r 

(2)  A  main  wave  C2D2,  giving  a  transmitted  wave  C/D/  and 

reflected  wave  A2B2. 

The  waves  moving  towards  the  transition  point  are  single  main 
waves,  AJ$i  and  C2D2,  and  the  waves  moving  away  from  the 
transition  point  are  combinations  of  waves  reflected  in  the  sec- 
tion and  waves  transmitted  from  the  other  section. 

69.  Considering  first  the  main  wave  moving  towards  rising  A: 
in  this  C2  =  0  =  D2,  hence,  from  (349), 


and 


(351) 


and  herefrom 


and 


C2    ~C1A 


-,+c, 


(352) 


REFLECTION  AND  REFRACTION  527 

which  substituted  in  (349)  gives 


and 


2c, 


-,  ,  1  . 

a,  a,  Cl  +  c 

Then  for  the  main  wave  in  section  1, 

=  £-v  £+«,A    ^  cog         _+B  sin 


(353) 


and 


sing  (A-  t)}. 


(354) 


When  reaching  a  transition  point  A  =  0,  the  wave  resolves  into 
the  reflected  wave,  turned  back  on  section  1,  thus: 


and 


(355) 


The  transmitted  wave,  which  by  passing  over  the  transition 
point  enters  section  2,  is  given  by 


and 


e   =  c 


(356) 


D 

The  reflection  angle,  tan  (t/)  =       ~  ,  is  supplementary  to  the 

B 

impact  angle,  tan  (ij  =  +  -~,  and  transmission  angle,  tan  (i'2) 


Reversing  the  sign  of  ^  in  the  equation  (355)  of  the  reflected 
wave,  that  is,  counting  the  distance  for  the  reflected  wave  also  in 
the  direction  of  its  propagation,  and  so  in  opposite  direction  as 


528  TRANSIENT  PHENOMENA 

in  the  main  wave  and  the  transmitted  wave,  equations  (355) 
become 


C2+C1 


(357) 


and  then 


or 

c, 


2      '      "1  V1J 


(358) 


(1)  In  a  single  electric  wave,  current  and  e.m.f.  are  in  phase 
with  each  other.    Phase  displacements  between  current  and 
e.m.f.  thus  can  occur  only  in  resultant  waves,  that  is,  in  the  com- 
bination of  the  main  and  the  reflected  wave,  and  then  are  a 
function  of  the  distance  ^,  as  the  two  waves  travel  in  opposite 
direction. 

(2)  When  reaching  a  transition  point,  a  wave  splits  up  into  a 
reflected  wave  and  a  transmitted  wave,  the  former  returning  in 
opposite  direction  over  the  same  section,  the  latter  entering  the 
adjoining  section  of  the  circuit. 

(3)  Reflection  and  transmission  occur  without  change  of  the 
phase  angle ;  that  is,  the  phase  of  the  current  and  of  the  voltage 
in  the  reflected  wave  and  in  the  transmitted  wave,  at  the  transi- 
tion point,  is  the  same  as  the  phase  of  the  main  wave  or  incoming 
wave.    Reflection  and  transmission  with  a  change  of  phase  angle 
can  occur  only  by  the  combination  of  two  waves  traveling  in 
opposite  direction  over  a  circuit;  that  is,  in  a  resultant  wave, 
but  not  in  a  single  wave. 

(4)  The  sum  of  the  transmitted  and  the  reflected  current 
equals  the  main  current,  when  considering  these  currents  in  their 
respective  direction  of  propagation. 


REFLECTION  AND  REFRACTION          529 

The  sum  of  the  voltage  of  the  main  wave  and  the  reflected 
wave  equals  the  voltage  of  the  transmitted  wave. 

The  sum  of  the  voltage  of  the  reflected  wave  and  the  voltage 
of  the  transmitted  wave  reduced  to  the  first  section  by  the  ratio 

g 

of  voltage  transformation  — ,  equals  the  voltage  of  the  main  wave. 

c\  c 

(5)  Therefore   a   voltage   transformation   by   the   factor    — 
ci 

IT     Q 

=  V  rT  7- occurs  at  the  transition  point;  that  is,  the  trans- 

T    C2     Li 

mitted  wave  of  voltage  equals  the  difference  between  main  wave 

£ 

and  reflected  wave  multiplied  by  the  transformation  ratio  —  : 

c  ci 

e2  =  —  (e^  —  e").    As  result  thereof,"  in  passing  from  one  section 

C2 

of  a  circuit  to  another  section,  the  voltage  of  the  wave  may 

^  . 

decrease  or  may  increase.  If  —  >  1,  that  is,  when  passing  from 

ci 

a  section  of  low  inductance  and  high  capacity  into  a  section  of 

high  inductance  and  low  capacity,  as  from  a  transmission  line 

into  a  transformer  or  a  reactive  coil,  the  voltage  of  the  wave  is 

s* 
increased;  if  —  <  1,  that  is,  when  passing  from  a  section  of  high 

ci 

inductance  and  low  capacity  into  a  section  of  low  inductance 
and  high  capacity,  as  from  a  transformer  to  a  transmission  line, 
the  voltage  of  the  wave  is  decreased. 

This  explains  the  frequent  increase  to  destructive  voltages, 
when  entering  a  station  from  the  transmission  line  or  cable,  of  an 
impulse  or  a  wave  which  in  the  transmission  line  is  of  relatively 
harmless  voltage. 

The  ratio  of  the  transmitted  to  the  reflected  wave  is  given  by 


2  VLjC,  2 


and 


2c2 


L2C, 


(359) 


530 


TRANSIENT  PHENOMENA 


60.  Example: 

Transmission  line 

Lt  =  1.95  X  1(T3 
Ct  =  0.0162  X  10-' 
ct  =  346 

^7,  =  0.56 
i* 

And  in  the  opposite  direction 


Transformer 


0.4  X  10-6 
1580 


.-?  "  2'56 


•J-    -0.56. 


The  ratio -^becomes  a  maximum,  =  GO,  for -1  =77,  but  in 

ei  ci      ^2 

this  case  e/'  =  0;  that  is,  no  reflection  occurs,  and  the  reflected 

wave  equals  zero,  the  transmitted  wave  equals  the  incoming 
wave. 


hence,  becomes  a  maximum  for  c2  =  0,  or  c^  =  oo  and 
then  =  2;  in  which  case  e2  =  0. 


2c, 


c2  + 
hence,  becomes  a  maximum  for  ct 


(360) 


0,  or  c2  =  oo  and  then  =  2; 


in  which  case  i2  =  0.  From  the  above  it  is  seen  that  the  maxi- 
mum value  to  which  the  voltage  can  build  up  at  a  single  transi- 
tion point  is  twice  the  voltage  of  the  incoming  wave,  and  this 
occurs  at  the  open  end  of  the  circuit,  or,  approximately,  at  a 
point  where  the  ratio  of  inductance  to  capacity  very  greatly 
increases. 


hence,  becomes  a  maximum,  and  equal  to  1,  for  cl  =  0, 
or  c2  =  oo. 

<  =  C2  ~  Cj 
«i       c2  +  ct 

has  the  same  value  as  the  current-ratio. 


(361) 


REFLECTION  AND  REFRACTION 


531 


61.  Consider  now  a  wave  traversing  the  circuit  in  opposite 
direction;  that  is,  C2D2  is  the  main  wave,  A ^B2  the  reflected 
wave,  C1D1  the  transmitted  wave,  and  Al=  0  =  B^  In  equa- 
tion (349)  this  gives 


C2  = 


and 
hence, 


ft  =  -ft 


and 


+ 

l    "I 


B,— Ac,--^!),.. 

J  ~  &  -  I  _  ^ 


D. 


(362) 


that  is,  the  same  relations  as  expressed  by  equations  (352)  and 
(353)  for  the  wave  traveling  in  opposite  direction. 

The  equations  of  the  components  of  the  wave  then  are  : 
Main  wave: 


2  cos  q 
{C2  cos  q 


sn  q 


D2  sin 


0  } 
0  }  ; 


(363) 


Transmitted  wave: 


sn 


sn 


^(364) 


Reflected  wave: 

i2  = if -«*«+*«*     {(72cosg  (>l  — /)— D2sin^  (^— 0} 


-  (365) 


532  TRANSIENT  PHENOMENA 

or,  in  the  direction  of  propagation,  that  is,  reversing  the  sign  of  ^ : 
1 If -**$-'»*  {C.  cos0  (/ +  0+^2  sing  (/  +  0  } 

I  (.*  i      \  r  ,£  J.      \  /     ) 


—  C. 


cos 


sn  q 


(366) 


62.   The  compound  wave,  that  is,  the  resultant  of  waves  pass- 
ing the  transition  point  in  both  directions,  then  is 


(367) 


4  &  &  &    s    j 

In  the  neighborhood  of  the  transition  point,  that  is,  for  values 
^  which  are  sufficiently  small,  so  that  e+8*  and  e~s*  can  be  dropped 
as  being  approximately  equal  to  1,  by  substituting  equations 
(354)  to  (356)  and  (363)  to  (366)  into  (367)  we  have 


=    ~Uot 


cos 


-  0  +  Bl  sin  q(l-t)}' 

'* {Al  cos  q  (A  +  0  -  #1  sin  q(l  +  t)} 

'1    +    C2 

'-^  {C2cosq  (J  +  0+  Da  sin  g(^  +  0}]; 
-Wo'  [{A^osg  (j  _  Q  +  J5tsing  (^  -  0 } 
'2  +  cx 

'l    "»      C2 

t  t   p 

~  \  C2  cos  q  (A  —  t)  —  D2  sin  q  (A  —  t) } 

'1    ~l~    C2 

2,c, 

-  { A  l  cos  ^  (^  —  t)  +  -Bj  sin  q  (^  —  0  }  ] 5 


(368) 


e°  = 


REFLECTION  AND  REFRACTION  533 

lot     [{C2  cos  q  (X  +  t)  +  D2  sin  q  (*  +  £)} 


cos 


-t    -        sn 


In  these  equations  the  first  term  is  the  main  wave,  the  second 
term  its  reflected  wave,  and  the  third  term  the  wave  transmitted 
from  the  adjoining  section  over  the  transition  point. 

Expanding  and  rearranging  equations  (368)  gives 

2  e  ~  w°' 
*!°  =  -f1— [{(c^!  -  caCJ  cos^  +  C2  (#1-  £2)  sin^}  cosqt 


-  { (clBl  +  c2D2)  cos  qX  -  c2(Al+  C2)  sin  qt}  sin  qt]; 


2c< 


2  s~Uot 


(369) 

ci  (Bi  -  #2)  sin  ql  }  cos  qt 

2 

-  {  (clBl  +  c2D2)  cos  ql-cl(Al  +  C2)  sin  g;}  sin  qt]; 


cos 


e2"  --  -    £    ^Ltci^i  +  ki;cos2x+^c1tf1  +  e2LJ2;sin2>ijcc 

Cl    +    C2 

-  { cl  (Bl  -  D2)  cos  qX  -  (c±A  l  -  c2C2)  sin  qX  \  si] 
63.  This  gives  the  distance  phase  angle  of  the  waves: 

tan  i  o  =     C2  {£i  ~  ^2)  c°s  $  +  (Al  +  C2)  sin  qt}   ^ 
(c^Ai  -  c2C2)  cos  qt  —  (clBl  +  c2D2)  sin  qt ' 

(c1Al-  c2C2)  cos  qt  -  (clBl  +  c2D2)  siuqt  ' 

no 


(370) 


tan  i2 


hence, 


tan^°      c2       T  L/2'ul 
(c.B.  +  c,D2)  cos  qt  +  (c.^  -  i 

fon    p    0    V     1  "* V     1         1 

tan  e,    —        (/^/^\  /T-»         T- 

c2{(Al  +  C2)  cos^  - 


c2(?2)  sin  ^ 
o  _  (cJS^  +  c2D2)  cos  $  -f  (CjAj  —  c2(72) 

I     tl     '     I         t/O  "  if          ^  ~A  ^         ^  ,     -^v  ^  ^ 


cos 


sin  9< 
-  D2)  sin  §( } 


(371) 
(372) 


534  TRANSIENT  PHENOMENA 

hence, 

tan62°_C          /LA. 


that  is,  at  a  transition  point  the  distance  phase  angle  of  the  wave 
changes  so  that  the  ratio  of  the  tangent  functions  of  the  phase 
angle  is  constant,  and  the  ratio  of  the  tangent  functions  of  the 
phase  angle  of  the  voltages  is  proportional,  of  the  currents 


inversely  proportional  to  the  circuit  constants  c  =  y  —  • 

In  other  words,  the  transition  of  an  electric  wave  or  impulse 
from  one  section  of  a  circuit  to  another  takes  place  at  a  constant 
ratio  of  the  tangent  functions  of  the  phase  angle,  which  ratio  is  a 
constant  of  the  circuit  sections  between  which  the  transition 
occurs. 

This  law  is  analogous  to  the  law  of  refraction  in  optics,  except 
that  in  the  electric  wave  it  is  the  ratio  of  the  tangent  functions, 
while  in  optics  it  is  the  ratio  of  the  sine  functions,  which  is  con- 
stant and  a  characteristic  of  the  media  between  which  the  tran- 
sition occurs. 

Therefore  this  law  may  be  called  the  law  of  refraction  of  a  wave 
at  the  boundary  between  two  circuits,  or  at  a  transition  point. 

The  law  of  refraction  of  an  electric  wave  at  the  boundary 
between  two  media,  that  is,  at  a  transition  point  between  two 
circuit  sections,  is  given  by  the  constancy  of  the  ratio  of  the 
tangent  functions  of  the  incoming  and  refracted  wave. 


CHAPTER  IX. 

INDUCTIVE  DISCHARGES. 

64.  The  discharge  of  an  inductance  into  a  transmission  line 
may  be  considered  as  an  illustration  of  the  phenomena  in  a 
complex  circuit  comprising  sections  of  very  different  constants; 
that  is,  a  combination  of  a  circuit  section  of  high  inductance  and 
small  resistance  and  negligible  capacity  and  conductance,  as  a 
generating  station,  with  a  circuit  of  distributed  capacity  and 
inductance,  as  a  transmission  line.  The  extreme  case  of  such  a 
discharge  would  occur  if  a  short  circuit  at  the  busbars  of  a  gen- 
erating station  opens  while  the  transmission  line  is  connected 
to  the  generating  station. 

Let  r  =  the  total  resistance  and  L  =  the  total  inductance  of 
the  inductive  section  of  the  circuit;  also  let  g  =  0,  C=  0,  and 
L0  =  inductance,  <70  =  capacity,  r0  =  resistance,  g0  =  conduc- 
tance of  the  total  transmission  line  connected  to  the  inductive 
circuit. 

In  either  of  the  two  circuit  sections  the  total  length  of  the 
section  is  chosen  as  unit  distance,  and,  translated  to  the  velocity 
measure,  the  length  of  the  transmission  line  is 


and  the  length  of  the  inductive  circuit  is 

~T  =  0;  (374) 


that  is,  the  inductive  section  of  zero  capacity  has  zero  length 
when  denoted  by  the  velocity  measure  X,  or  is  a  "massed  induc- 
tance." 

It  follows  herefrom  that  throughout  the  entire  inductive 
section  X  =  0,  and  current  i1  therefore  is  constant  throughout  this 
section. 

Choosing  now  the  transition  point  between  the  inductance  and 
the  transmission  line  as  zero  of  distance,  A  =  0,  the  inductance 

635 


536 


TRANSIENT  PHENOMENA 


is  massed  at  point  ^  =  0,  and  the  transmission  line  extends  from 
X  =  0  to  X  =  V 

Denoting  the  constants  of  the  inductive  section  by  index  1, 
those  of  the  transmission  line  by  index  2,  the  equations  of  the  two 
circuit  sections,  from  (290),  are 


Cj)  cos  qt  — 


-(51  +  7)1)sin^},l 
-  A-DJsing*};  j 


(376) 


i2  =  £-Uot{s  +sA  [A2  cos  q  (X  -  t)  +  B2  sin  g  (^  -  t)] 
—  s~s*  [C9  cos  q  (A  +  t)  +  Z),  sin  0  (^  +  £)"]}. 

L        /  Z\  /  ^  1\  x-!)^ 

e2  =  ca  e-rt  {[£+^  [A2  cos  g  (^  -  0  +  J52  sin  q  (I  -  t)] 

+  £~sA  [Ca  cos  g  (^  +  0  +  D2  sin  q  (Jl  +  0]}, 

and  the  constants  of  the  second  section  are  related  on  those  of 
the  first  section  by  the  equations  (285) : 

A        -  n    A      4-   7)  n  C1 

Aj    -     «!/!.!    "I'    Oil/ti  ^2 

D  „    D  ^\    T\  T\ 

•D  n      ==     U/fJJ-i      t/1iyi.  -^'•5 

where 

«,=VC-' 

(286) 


+ 


(349) 


and 


(287) 


65.  In  the  inductive  section  having  the  constants  L  and  r, 
that  is,  at  the  point  A  =  0  of  the  circuit,  current  il  and  voltage 
et  must  be  related  by  the  equation  of  inductance, 


(377) 


Substituting  (375)  in  (377),  and  expanding,  gives 
ci  { (Ai  +  CJ  cos  qt  -  (B^  -  DJ  sin  qt] 

=  (r  +  w0L)  { (At  -  C,)  cos  qt  -  (B^  +  D,)  sin  qt} 
+  ^L  { (Al  -  CJ  sin  ^  +  (B1  +  DJ  cos  qt}, 


INDUCTIVE  DISCHARGES 

and  herefrom  the  identities 

Cl  (A,  +  C,)  =  (r  +  ^)  (^i  ~  Ci)  +  ^  (*i  +  *>i), 
Cl  (B,-D,)  =  (r  +  u0L)  (5,  +  ig  -  gL  (Ax-  CJ. 

Writing 
and 


gives 

cx  (AA  +  CJ  =  (r  +  w0^)  M  +  3LN    ] 

and  | 

c,  (5,  -  Dx)  =  (r  +  w0L)  A^  -  gLM,  J 

which  substituted  in  (349)  gives 

A2  =  —  { (c  +  r  +  w0£)  ^  +  ^^ }  ^  -  (M  +  p^V), 

J52  =  1  { (c  +  r  +  u0L)  N  -  qLM}  ~*:(N  -  pM), 
z  c  £ 


537 


(378) 


(379) 


(380) 


- 

Z  C 


-  (c  -  r  - 


-f  (c  -  r-  m0L) 


-  M), 


JV), 


(381) 


where  in  the  second  expression  terms  of  secondary  order  have 
been  dropped. 


P 


qL 


Then   substituting  in    (375)    gives    the   equations    of   massed 
inductance : 

it  =  e  ~M  { M  cos  qt  -  N  sin  qt } 


(382) 

If  at  t  =  0,  £j  =  0,  that  is,  if  at  the  beginning  of  the  transient 
discharge  the  voltage  at  the  inductance  is  zero,  as  for  instance 
the  inductance  had  been  short-circuited,  then,  substituting  in 


538 


TRANSIENT  PHENOMENA 


(382),  and  denoting  by  i0  the  current  at  the  moment  t  =  0,  or  at 
the  moment  of  start,  we  have 


t  =  0,  i\=  %,«!  =  0;  hence, 
M  =  i 


(383) 


and 


r  T  -}-  u  Li 

h  =  V~"u*  1  cos  <#  +        -r2"  sm  ^ 

gL 

+  (r  +  V')2  , 


(384) 


In  this  case 

i 
A*=2' 


2  +  (r  +  uQL)2  +  c  (r 

\  -  \J          s      f     / 

(385) 


+  (r  +  ^0L)2  -  c  (r  + 


66.   In  the  case  that  the  transmission  line  is  open  at  its  end, 
at  point  A  =  ^0, 

>l   =  1 


?•-*-?' 

i. 


hence,  this  substituted  in  (376),  expanded  and  rearranged  as 
function  of  cos  qt  and  sin  qt,  gives  the  two  identities 

£+sA°  (A,  cos  g;o+  B2  sin  gao)  «  fi-^  (C2  cos  q*0  +  D2  sin  ^, 

and 

c+«fc// 


(386) 


Squared  and  added  these  two  equations  (386)  give 
(A?  +  £2)  .  ,-2^o   C2  +  D  2> 


(387) 


INDUCTIVE  DISCHARGES 


539 


Divided  by  each  other  and  expanded  equations   (386)  give 

(Af,  -  B2DJ  sin  2  gA0  -  (A2D2  +  B2C2)  cos  2  qXQ.    (388) 

Substituting  (381)  into  equations  (387)  and  (388)  gives 

2^{(qL)2  +  (c  +  r  +  u0L)2}=e-*s**{(qL)2+(c-r-uQL)2}    (389) 

{  (qL)2  +  (r  +  u,L)2  -  c2  \  sin  2  qXQ  =  2  cgL  cos  2  ^0.   (390) 

Since  2  s^0  is  a  small  quantity,  in  equation  (389)  we  can  sub- 


stitute 


hence,  rearranging  (389)  and  substituting 

s  =  u0  —  u 
gives 

c  (r  +  u0L)  -  (u  -  u0)  ;0  { (qL)2+(r  +  u0L)2  +  c2 }  =  0.    (391) 

Since  (r  +  u0L)  is  a  small  quantity  compared  with  c2  (qL)2,  it 

can  be  neglected,  and  equations  (390)  and  (391)  assume  the  form 

{ (qL)2  -  c2 }  sin  2  gA0  =  2  cqL  cos  2  q*Q  (392) 

c  (r  +  u,L)  -  (u  -  UQ)  *0  { (qL)2  +  c2}  =  0,         (393) 

and,  transformed,  equation  (392)  assumes  the  form 

2cqL 


tan  2 


or 


or 


(qL)2  -  c2  ' 


q  =  -    -  tan  ql0, 


(394) 


q  =  +  -f  cot  g^0, 
L 

hence  tan  2  q^  is  positive  if  #L  >  c,  as  is  usually  the  case. 
Expanded  for  u0,  equation  (393)  assumes  the  form 


+cL 


or 


(L+r-) 
V         wl 


u       cL  +  ^0  { (qL)2  +  c2 } 
s  =  -  (u  -  u0). 


(395) 


540 


TRANSIENT  PHENOMENA 


From  equations  (394)  q  is  calculated  by  approximation,  and 
then  from  (395)  u0  and  s. 

As  seen,  in  all  these  expressions  of  q,  uw  s,  etc.,  the  integration 
constants  M  and  N  eliminate;  that  is,  the  frequency,  time  atten- 
uation constant,  power  transfer,  etc.,  depend  on  the  circuit  con- 
stants only,  but  not  on  the  distribution  of  current  and  voltage 
in  the  circuit. 

67.  At  any  point  X  of  the  circuit,  the  voltage  is  given  by  equa- 
tion (376),  which,  transposed,  gives 

e  =  c£-wo<{£+^  [(42  cos  q%  +  %2  sin  qX)  cos  qt 
+  (A2  sin  qX  —  B2  cos  qX)  sin  qt] 
+  e~sA  [(C2  cos  qX  +  D2  sin  qX)  cos  qt 
—  (C2  sin  qX  —  D2  cos  qX)  sin  qt] } , 

or  approximately, 

e  =  cs~u<*  {[(A2  +  C2)  cos  qX  +  (B2  +  D2)  sin  qX]  cos  qt 
+  [(A2  -  C2)  sin  g^  -  (B2  -  D2)  cos  ^]  sin  qt}. 

Similarly  to  equation  (381), 

A2  +  C2  =  pN', 
A2-C2  =  M; 
B*  +  D2  =  N; 


(396) 


where 
then 


T)  — • 


cos  ^A+c  sin  gd)  (JV  cos  ^  +  M  sin  ^), 


(cos  ^  _       sin 


cos  g«  -  N  sin  $); 


^1  (N  cos  qt  +  M  sin  $), 
(M  cos  qt-N  sin  g«). 


(397) 
(398) 


INDUCTIVE  DISCHARGES 


541 


If 


hence, 


e,  =  0   for  t  =  0, 
N  =  0; 

h  =  V  ~^  cos  $> 


'2       "      0 


,    <1L  •    n 

(cos  qX  -    -  sin  g/)  cos 
c 


c  sn 


Writing 


(M2 


the  effective  values  of  the  quantities  are 


I2  = 


E2  =  I0e~u<*  (qL  cos  q\  +  c  sin  gA).  . 

Herefrom  it  follows  that 

/2  =  0  for  A  =  ^0  by  the  equation 

cos  aAn  -  q- sin  gA0  =  0, 


or 

while 
gives 

that  is, 


q  -  zco 


«-  - 


^L  cos  ql0  +  c  sin  gA0  =  0; 


(399) 


(400) 


(401) 


(402) 
(403) 


0  at 


542 


TRANSIENT  PHENOMENA 


At  the  open  end  of  the  line  ^  = 
tuting  (402)  into  (401)  is 


the  voltage  E2  by  substi- 


cos 


If 


+ 


At  the  grounded  end  of  the  line  X  = 
stituting  (403)  into  (401),  is 

7  ~  —  U(jt, 

V 


(404) 
the  current  72,  by  sub- 


(405) 


An  inductance  discharging  into  the  transmission  line  thus 
gives  an  oscillatory  distribution  of  voltage  and  current  along  the 
line. 

68.  As  example  may  be  considered  the  three-phase  high- 
potential  circuit,  comprising  a  generating  system  of  r  =  2  ohms 
and  L  =  0.5  henry  per  phase  and  connected  to  a  long-distance 
transmission  line  of  r0=  0.4  ohm,  L0  =  0.002  henry,  gQ=  0.2  X 
10~6  mho,  (70=  0.016  X  10~6  farad  per  mile  of  conductor  or 
phase,  and  of  1Q  =  80  miles  length. 


c  =\/-  = 


c2  =  125,300; 


<TO  =  VL0C0  -  5.66  X  10-6; 
^o  =  Zo°"o  =  0.453  X  10-3; 
l/rn 


L  =  7°8' 


and  herefrom,  substituting  in  equations  (394)  and  (395), 
q  =  -  708  tan  (0.0259  q)°     (zero  voltage) 
=  +  708  cot  (0.0259  q)°     (zero  current), 


u      0.618  (f  10-6  +  1.28 


H  = 

100.35° 

185.64° 

273.83° 

362.89° 

452.32° 

541.94° 

?  = 

3875 

7168 

10,572 

14,010 

17,463 

20,920 

u 

0.0946 

0.0302 

0.0142 

0.00816 

0.0047 

0.0037 

W0  = 

95.8   . 

102.8 

104.5 

105.1 

105.5 

105.6 

jr- 

10.2 

3.2 

1.5 

0.87 

0.5 

0.4 

INDUCTIVE  DISCHARGES  543 

By  equation  (401)  the  effective  values  of  the  first  six  har- 
monics are  given  as 

(1)  Quarter-wave:  100.35°. 

q,  =  3875; 
u0  =  95.8; 

7  =  if-"*  (cos  qX  -  5.48  sin  qX)', 
E  ==  1939  v'^  (cos  qX  +  0.182  sin  qX). 

(2)  Half-wave:  185.64°. 

q2  =  7168; 
UQ  =  102.8; 

7  =  if-"*  (cos  qX  -  10.14  sin  qX); 
E  =  3585  iQr^  (cos  ^  +  0.098  sin  qX). 

(3)  Three-quarter  wave:  273.83°. 

qz  --  10,572; 
u0  =  104.5; 

7  =  v~^  (cos  g^  -  14.90  sin  qX); 
E  =  5287  i0e-Uot  (cos  ^>i  +  0.067  sin  qX). 

(4)  Full  wave:  362.89°. 

?4  ••  14,010; 
w0  =  105.1; 

7  =  v~^  (cos  qX  -  19.8  sin  qX); 
E  =  7005  v~^  (cos  qX  +  0.050  sin  qX). 

(5)  Five-quarter  wave:  452.32°. 

<?5  »  17,463; 
w0  =  105.5; 

7  =  v'7*0'  (cos  qX  -  24.65  sin  gfl; 
E  =  8732 1>-^  (cos  qX  +  0.040  sin  gj). 


544  TRANSIENT  PHENOMENA 

(6)  Three-half  wave:  541.94°. 

&  =  20,920; 
UQ  =  105.6; 

7  =  if-**  (cos  qX-  29.6  sin  gd); 
jE/  =  10,460  v~Wo<  (cos  g>l  +  0.033  sin  qX). 


APPENDIX 


VELOCITY  FUNCTIONS  OF  THE  ELECTRIC  FIELD 

IN  the  study  of  the  propagation  of  the  electric  field  through 
space  (wireless  telegraphy  and  telephony),  a  number  of  new 
functions  appear  (Section  III,  Chapter  VIII). 

.  By  the  following  equations  these  functions  are  defined,  and 
related  to  the  "  Sine-Integral"  Si  x,  the  "  Cosine-Integral"  Ci  x, 
and  the  "  Exponential  Integral,"  Ei  x,  of  which  tables  were 
calculated  by  J.  W.  L.  Glaisher  (Philosophical  Transactions  of 
the  Royal  Society  of  London,  1870,  Vol.  160)  : 


col  x  =  —  du 

u*    u5    u7  du 


l_v?     }_u?     l 

3\3+55~7 


|2     [4     |6 
=  cosxj--^+p-p+-...j 

fl      |3     |S     |7 
H-sinz    -§-^  +  ^-^5 

I 


71       /^  sin  u 
2"Jo       « 


du. 

545 


546 

sil  x=  I  du 


APPENDIX 


rx  cos  u 

ll  x=      ~>1~( 

Jx 

C*\i--  ---       1-^ 


1  u2     1  u*    1 


IX4 


l  |3  |5  |7 
—  -^+i=-^ 
a;2  a^  x6  x8 


l      2     |4     |5 
--%4^-ty 

X      X3      X5      X7 


fx 
^du; 


where 


r=  0.5772156  .  .  . 
/•<?  t*          /•«   _„ 

Explx=        -dt*=  (      — 
Jx   u         J_x   u 


u 


du 
u 


1  M2      1  M3      1 


/« 


where 


l     1       2 

J 


r-x^ 
J«     ~v^du' 


x4    x5 


q  is  given  by :  expl  q  =  0  as :     q  =  0.37249680  . 


APPENDIX 


547 


Expl  ( 


f»     -u 
—du 
U 


du 


/.  1  u2     1  u3     1  u4 

=  /logtt— u  +  2  T^- Q  1^+4  17-  +•  •  • 

/  ^    |^S         O    |O         T:    |T: 


1  x2    1  x3    1 14 


du. 


Tables  of  these  four  functions,  reduced  from  the  Glaisher 
tables,  are  given  in  the  following  for  6  decimals,  and  their 
first  part  plotted  in  Fig.  103. 

These  functions  have  the  following  properties: 


col  0  =  ^ 

sil  0  •-•=  +  oc 
expl  0  =  +  oo 
expl  q  =  0 


col 


=  0 


sil  oo  =0 

expl  ( +  oo  )  =  -  oo 
expl  (—  co)=0 


col  x  has  maxima  at  the  even,  sil  x  at  the  odd  quadrants, 
and  these  maxima  are  alternately  positive  and  negative;  that  is, 


col  ^2s          =max. 


sil  o(2s—  l)=max. 


=  integer  number. 


548 


APPENDIX 


.5       1.0      1.5      2.0     2.5      3.0     3.5      4.0     4.5     5.0      5.5      6.0      6.5      7.0     7.5 

BIG.  103. 


APPENDIX  549 

For  large  values  of  s,  the  numerical  values  of  these  maxima 
approach  the  values: 


for:  s>40,  the  approximation  is  correct  to  the  6th  decimal. 
For  small  values  of  x,  the  approximations  hold : 

si\x  =  log-  -0.5772156, 


For  large  values  of  x,  it  is  : 

2  sin  x  2  sin  x 


2  cos  x  2  cos 

-  <  col  x  <  — 


where  n\  and  n2  are  the  two  successive  quadrants,  between 
which  x  lies.     For  instance  : 


sin  x  =  sin  1.5^"=  0.707, 


cos  x  =  cos  1.5^-=  —0.707, 


550  APPENDIX 

and  0.01 725  <sil  x<  0.01805, 

-0.01805  <  col  x<  -0.01725. 
It  is,  approximately,  for  large  values  of  x,  if : 


col  x  =  co\  —ucos  v, 


sil  x  =  sil  pr-ucos  v, 


where 


and  u  is  the  nearest  even  quadrant  to  x,  for  col  x,  and  the  nearest 

odd  quadrant  to  x,  for  sil  x,  and  col  ^  u  or  sil  ^  u  are  given  by 
Table  VII. 

For  instance  : 


vi=  -0.785  v2=  +0.785 

col  26  £  =  -0.0245  sil  25  £  =  0.0254 

_  l 

'cos  (-0.785)  =0.707  cos  0.785  =  0.707 

col  40  =  -0.01730  sil  40  -  0.01795 

From  the  series  expressions  of  these  functions  follows: 
col  (  —  x)     =n  —  col  x 
sil  (—  x)      =sil  x 
expl  (jx)     =  sil  x  +  y  col  x  -  / 


APPENDIX  551 

.  7T 

expl  (  —  jx)=su  x—]  col  £  +  7  o 

£ 

col  (±jx)    =  ±|iexpl  x-expl  (  —  x)\  +/TT 

1 

sil  (±jx)     =^-{expl  x  +  expl  (  —  x)} 


552 


APPENDIX 


TABLE  I 

Col  x  and  sil  x  from  0.00  to  1.00 


X 

cola; 

J 

sil  x 

J 

.00 

+  1  .  570  796 

in  onn 

GO 

.01 
.02 
.03 
.04 

1.560  796 
1  .  550  797 
1.540  798 
1  .  530  800 

9  999 
9  999 
9  998 

Q  QQ7 

+  4.027  980 
3.334  907 
2.929  567 
2.642  060 

693  073 
405  340 

287  507 

999  QIC 

ftK 

1  520  803 

2  419  149 

zzz  yio 

9QQC 

.06 
.07 
.08 
.09 

1.510  808 
1.500  815 
1.490825 
1.480837 

9  993 
9  990 

9  988 

9QO  r 

2.237  095 
2.083  269 
1.950  113 
1.832  754 

loZ  U4/ 
153  826 
133  156 
117  359 

.10 

1.470  852 

yoo 

1  .  727  868 

104  886 

.11 
.12 
.13 
.14 

1.460  870 
1.450  892 
1.440  918 
1.430  949 

9  982 
9  978 
9  974 
9  969 

9QA  K 

1.633  083 
1  .  546  646 
1.467  227 
1.393  793 

94  785 
86  437 
79  419 
73  434 

.15 

1.420  984 

1.325  524 

68  269 

.16 
.17 
.18 
.19 

1.411  024 
1.401  069 
1.391  120 
1.381  177 

9  960 
9  955 
9  949 
9  943 

9QQ7 

1.261  759 
1.201  957 
1.145  672 
1.092  527 

63  765 
59  802 
56  285 
53  145 

.20 

1.371  240 

\)6i 

1.042  206 

50  321 

.21 
.22 
.23 
.24 

1.361  310 
1.351  387 
1.341  471 
1.331  563 

9  930 
9  923 
9  916 
9  908  ' 
9onn 

.994  437 
.948  988 
.905  656 
.864  266 

47  769 
45  449 
43  332 
41  390 

.25 

1.321  663 

yuu 

.824  663 

39  603 

.26 
.27 

.28 
.29 

1.311  771 
1.301  888 
1.292  013 

1.282  148 

9  892 
9  883 
9  875 
9  865 

.786  710 
.750  287 
.715  286 
.681  610 

37  953 
36  423 
35  001 
33  676 

.30 

1.272  292 

006 

.649  173 

32  437 

.31 
.32 
.33 
.34 

1.262  447 
1.252  611 
1.242  786 
1.232  972 

9  845 
9  836 
9  825 
9  814 

.617  896 
.587  710 
.558  549 
.530  355 

31  277 
30  186 
29  161 
28  194 

.35 

1.223  169 

803 

.503  076 

27  279 

.36 
.37 

.38 
.39 

1.213  378 
1.203  599 
1.193  832 
1  .  184  077 

9  791 
9  779 
9  767 
9  755 

9*7,49 

.476  661 
.451  067 
.426  252 
.402  178 

26  414 
25  594 
24  815 
24  074 

.40 

1.174  335 

/4Z 

.378  809 

23  369 

.41 
.42 
.43 
.44 

1  .  164  606 
1.154  891 
1.145  189 
1.135  501 

9  729 
9  715 
9  702 
9  688 

.356  114 
.334  062 
.312  625 
.291  776 

22  695 
22  052 
21  437 
20849 

.45 

1  .  125  828 

O/o 

.271  492 

20  284 

.46 
.47 

.48 
.49 

1.116  170 
1.106  526 
1.096  898 
1.087  285 

9  658 
9  644 
9  628 
9  613 

.251  749 
.232  526 
.213  803 
.195  562 

19  743 
19  223 
18  723 
18  241 

.50 

+  1  .  077  689 

_  O.M) 

+  0.177  784 

17  778 

APPENDIX 
TABLE   I— Continued 


553 


X 

col  x 

J 

sil  x 

J 

50 

+  1  077  689 

9  596 

+  0  177  784 

17  778 

9C01 

1  7  oo-i 

.51 
.52 
.53 
.54 

1.068  108 
1.058  545 
1.048998 
1.039448 

9  563 
9  547 
9  530 

9r  1  9 

.160453 
.143  554 
.127  071 
.110  990 

11  OOl 

16  899 
16  483 
16  081 
i  £  AQH 

55 

1  029  956 

Ol/ 

095  300 

lo  oyu 

9JOK 

i  ^  *?i  d. 

.56 
.57 

.58 
.59 

1.020  461 
1.010  985 
1.001  527 
.992  088 

9476 
9  458 
9439 

94.91 

.079  986 
.065  037 
.050  441 
.036  190 

14  949 
14  596 
14  251 

1  Q  Q1  Q 

60 

982  667 

022  271 

9AC\1 

.61 
.62 
.63 
.64 

.973  266 
.963  885 
.954  523 
.945  182 

4U1 

9  381 
9  362 
9  341 
9  001 

+  .008  675 
-  .004  e06 
-  .017  582 
-  .030  260 

lo  olo 
13  281 
12  976 
12  678 

1  9  QQfl 

.65 

.935  861 

641 

-  .042  650 

iz  oyu 

.66 
.67 
.68 
.69 

.926  561 
.917  282 
.908  024 

.898  788 

9  300 
9  279 

9  258 
9  236 

991  J. 

-  .054  758 
-  .066  591 
-  .078  158 
-  .089  463 

12  108 
11  833 
11  567 
11  305 

nnco 

.70 

.889  574 

-  .100  515 

.71 
.72 
.73 

.74 

.880  382 
.871  213 
.862  066 
.852  942 

9  192 
9  169 
9  147 
9  124 
91  on 

-  .111  318 
-  .121  879 
-  .132  203 
-  .142  296 

10  803 
10  561 
10  324 
10  093 

9QAQ 

75 

843  842 

luu 

—  152  164 

oOo 

9fY77 

9RAK 

.76 

.77 
.78 
.79 

.834  765 
.825  713 
.816  684 
.807  680 

9  052 
9  029 
9  004 
8osn 

-  .161  810 
-  .171  240 
-  .180  458 
-  .  189  470 

O4O 

9  430 
9  218 
9  012 
8  s  no 

.80 

.798  700 

you 

-  .198279 

ouy 

.81 
.82 
.83 
.84 

.789  746 
.780  817 
.771  913 
.763  035 

8  954 

8929 
8904 

8  878 

80  r  -I 

-  .206  889 
-  .215  305 
-  .223  530 
-  .231  568 

8  601 
8416 
8  225 
8  038 

85 

754  184 

OOl 

239  423 

OOO 

8  one 

7fi7  ^ 

.86 

.87 
.88 
.89 

.745  358 
.736  560 

.727  788 
.719  043 

8  798 
8  772 
8  745 

871  Q 

-  .247  098 
-  .254  597 
-  .261  923 
-  .269  079 

7  499 
7  326 
7  156 

6QQQ 

.90 

.710  325 

-  .276  068 

.91 
.92 
.93 
.94 

.701  636 
.692  974 
.684  340 
.675  735 

8  689 
8  662 
8  634 
8  605 

8K77 

-  .282  893 
-  .289  558 
-  .296  064 
.302  415 

6  825 
6  665 
6  506 
6  351 

61  QQ 

.95 

.667  158 

.30S  614 

iyy 

.96 

.97 
.98 
.99 

.658  610 
.650  092 
.641  602 
.633  143 

8  548 
8  518 
8  490 
8  459 

8    A  Of\ 

-  .314  662 
-  .320  563 
-  .326  319 
-  .331  931 

6  048 
5  901 
5  756 
5612 

5170 

1.00 

+  0.624  713 

4oU 

-0.337  404 

•1M 

554 


APPENDIX 


TABLE   II 

Expl  x  and  expl  (-x)  from  0.00  to  1.00 


X 

expl  x 

J 

expl  (-a;) 

A 

.00 

+  GO 

00 

.01 
.02 
.03 
.04 

+  4.017  929 
3.314  707 
2.899  116 
2.601  257 

703  223 
415  591 

297  859 

+  4.037  930 
3.354  708 
2.959  119 
2.681  264 

684  222 
395  589 
277  855 

05 

2  367  885 

Zoo  otZ 

2  467  898 

213  366 

.06 
.07 
.08 
.09 

2.175  283 
2.010  800 
1.886  884 
1.738  664 

JL\)A  DU^ 

164  483 
143  916 
128  220 

2.295  307 
2.150  838 
2.026  941 
1.918  745 

i/z  oyl 
144  469 
123  897 
108  196 

10 

1  622  813 

llo  ool 

1  822  924 

95  821 

1  f\  C   0  CA 

.11 

.12 
.13 
.14 

1.516  959 
1.419  350 
1.328655 
1.243  841 

97  609 
90695 

84  814 

1.737  107 
1.659  542 
1.588  899 
1.524  146 

oO  ol/ 

77  565 
70643 
64  753 

15 

1  164  086 

/y  i  o4 

1  464  462 

59  684 

7  e  o  re 

.16 
.17 
.18 
.19 

1.088  731 
1.017  234 
.949  148 
.884  096 

to  oOo 
71  497 

68  087 
65  052 

1.409  187 
1.357  781 
1.309  796 
1.264  858 

oo  ^/o 
51  406 

47  985 
44  938 

.20 

.821  761 

o^  ooo 

1.222  651 

42  207 

.21 
.22 
.23 
.24 

.761  872 
.704  195 
.648  529 
.594  697 

59  889 
57  676 
55  666 
53  832' 

1.182  902 
1.145  380 
1.109  883 
1.076  235 

39  749 
37  522 
35  497 
33  648 

.25 

.542  543 

QL  loo 

1.044  283 

31  952 

.26 
.27 
.28 
.29 

.491  932 
.442  741 
.394  863 
.348  202 

50  611 
49  191 

47  878 
46  662 

1.013  889 
.984  933 
.957  308 
.930  918 

30  394 
28  956 
27  625 
26  390 

.30 

.302  669 

4o  Ooo 

.905  677 

25  241 

.31 
.32 
.33 
.34 

.258  186 
.214  683 
.172  095 
.  130  363 

44  482 
43  503 
42  588 
41  732 

.881  506 
.858  335 
.836  101 
.814  746 

.24  171 
23  171 
22  234 
21  355 

.35 

.089  434 

4u  y^y 

.794  215 

20  531 

.36 
.37 
.38 
.39 

.049  258 
+  .009  790 
-  .029  Oil 
-  .067  185 

40  176 

39  468 
38  801 
38  173 

.774  462 
.755  441 
.737  112 
.719437 

19  753 
19  021 
18  329 
17  675 

.40 

-  .104  765 

61  Ool 

.702  380 

17  057 

.41 
.42 
.43 
.44 
45 

-  .141  786 
-  .178278 
-  .214  270 
-  .249  787 
284  855 

37  021 
36492 
35  991 
35  517 
35  068 

.685  910 
.669  997 
.654  613 
.639  733 

fl'JK  '-JQ1 

16  470 
15  913 
15  384 
14  880 
14  402 

.46 
.47 
.48 
.49 

.319  497 
-  .353  735 

-  .387  589 
-  .421  078 

34  642 
34  238 
33  854 
33  489 

.611  387 
.  597  877 
.584  784 
.572  089 

13  944 
13  510 
13  093 
12  695 

.50 

-0.454  220 

33  142 

+  0.559  774 

12  315 

APPENDIX 
TABLE   II— Continued 


555 


X 

expl  x 

J 

expl  (  —  x) 

A 

.50 

-0.454  220 

33  142 

+  0.559  774 

12  315 

.51 

.52 
.53 
.54 

-  .487  032 
-  .519  531 
-  .551  730 
-  .583  646 

32  812 

32  498 
32  200 
31  915 

o-i  nA  r 

.547  822 
.536  220 
.524  952 
.514  004 

11  952 
11  602 
11  268 
10  948 

.55 

-  .615  291 

.503  364 

10  640 

.56 
.57 

.58 
.59 

-  .646  677 
-  .677  819 
-  .708  726 
-  .739  410 

31  387 
31  141 
30  907 
30  684 

.493  020 
.482  960 
.473  173 
.463  650 

10  344 
10  060 

9  787 
9  523 

.60 

-  .769  881 

6(J  QiZ 

.454  380 

9  270 

.61 
.62 
.63 
.64 

-  .800  150 
-  .830  226 
-  .860  119 
-  .889  836 

30  269 
30  076 
29  892 
29  717 
90  £^n 

.445  353 
.436  562 
.427  997 
.419  652 

9  027 
8  791 
8  565 
8  345 

.65 

-  .919  386 

zy  oou 

.411  517 

135 

.66 
.67 
.68 
.69 

-  .948  778 
-  .978  019 
-1.007  116 
-1.036  077 

29  392 
29  241 
29  079 
28  960 

.403  586 
.395  853 
.388  309 
.380  950 

7  931 
7  733 
7  544 
7  359 

.70 

-1.064  907 

28  80  1 

.373  769 

7  181 

.71 
.72 
.73 
.74 

-1.093  615 
-1.122  205 
-1.150  684 
-1.179  058 

28  707 
28  590 
28  479 
28  374 

.366  760 
.359  918 
.353  237 
.346  713 

7  009 
6  842 
6  681 
6  524 

.75 

-1.207  333 

Zo  Z.  O 

.340  341 

6  3,2 

.76 

.77 
.78 
.79 

-1.235  513 
-1.263  605 
-1.291  613 
-1.319  542 

28  181 
28  092 
28  008 
27  929 

97  Q££ 

.334  115 
.328  032 
.322  088 
.316  277 

6  226 
6  083 
5  944 
5  811 

.80 

1.347  397 

Z<  OOO 

.310  597 

5  680 

.81 

.82 
.83 

.84 

-1.375  182 
-1.402  902 
-1.430  561 
-1.458  164 

27  785 
27  720 
27  659 
27  603 

.305  043 
.299  611 
.294  299 
.289  103 

5  554 
5  432 
5  312 
5  196 

.85 

-1.485  714 

27  ooU 

.284  019 

5  084 

.86 

.87 
.88 
.89 

-1.513  216 
-  1  .  540  673 
-  1  .  568  089 
-  1  .  595  467 

27  502 
27  457 
27  416 
27  379 

97  QA  £ 

.279  045 
.274  177 
.269  413 
.264  749 

4  974 
4  868 
4  764 
4  664 

.90 

-1.622  812 

£i  O4O 

.260  184 

565 

.91 
.92 
.93 
.94 

-1.650  126 
-1.677  413 
-1.704  677 
-1.731  920 

27  314 

27  287 
27  264 
27  243 

97  99R 

.255  714 
.251  336 
.247  050 
.242  851 

4  470 
4  378 

4  286 
4  199 

95 

—  1  759  146 

238  738 

llo 

97  91  1 

.96 
.97 
.98 
.99 

-1.786  357 
-1.813  557 
-1.840  749 
-1.867  935 

Z/  ZLL 

27  200 

27  192 
27  186 

97  1  QQ 

.234  708 
.230  760 
.226  891 
.223  100 

UoU 

3  948 
3  869 
3  791 

1.00 

-1.895  118 

Zt  loo 

+  0.219  384 

/  lo 

556 


APPENDIX 


TABLE   III 

Col  x  and  sil  x  from  0.0  to  5.0 


X 

col  x 

J 

sil  x 

A 

.0 

+  1.570  796 

oo 

.1 
.2 
.3 
.4 

1.470  852 
1.371  240 
1.272  292 
1.174  335 

+  1.727  868 
1.042  206 
.649  173 
.378  809 

1  H77  ASQ 

mnoA 

.6 

.7 
.8 
.9 

.982  667 
.889  574 
.798  700 
.710  325 

+  .022  271 
-  .100  515 
-  .198  279 
-  .276  068 

10 

R94.  71  "^ 

007  A(\A 

_i_  09  cno 

1.1 
1.2 
1.3 
1.4 

.542  111 
.462  749 

.387  838 
.314  570 

79  362 
75  911 

72  269 

f»o  A  cn 

-  .384  873 
-  .420  459 
-  .445  739 
-  .462  007 

+  4/  4by 
35  586 
25  279 
16  -268 

1.5 

.246  113 

Do  4O/ 

-  .470  356 

o4y 

1.6 
1.7' 

1.8 
1.9 

.181  616 
.121  204 
.064  979 
+  .013  021 

64  497 
60412 
56  225 
51  958 

A  7  ftQQ 

-  .471  733 
-  .466  968 
-  .456  811 
-  .441  940 

+  1  377 
-  4  765 
-10  157 

-14  871 

2.0 

-  .034  617 

*±l  DOo 

-  .422  981 

—  18  959 

2.1 
2.2 
2.3 
2.4 

-  .077  902 
-  .116  839 
-  .151  411 

-  .18  689 

43  285 
38  927 
34  582 
30  278  ' 

-  .400  512 
-  .375  075 
-  .347  176 
-  .317  292 

-22  469 
-  25  437 

-27  899 
-29  884 

2.5 

-  .207  724 

Zo  Ooo 

-  .285  871 

—  31  421 

2.6 
2.7 

2.8 
2.9 

-  .229  598 
-  .247  416 
-  .261  300 
-  .271  394 

21  874 
17  818 
13  884 
10  094 

-  .253  337 
-  .220  085 
-  .  186  488 
-  .152  895 

-32  534 
-33  252 
-33  597 
-33  593 

3.0 

-  .277  856 

4oZ 

-  .119  630 

—  33  265 

3.1 
3.2 
3.3 
3.4 

-  .280  863 
-  .280  605 
-  .277  284 
-  .271  118 

+  3  007 
258 
-  3  319 
-  6  166 

-  .086  992 
-  .055  257 
-  .024  678 
+  .004  518 

-32  638 
-31  735 
-30  579 
-29  196 

3.5 

-  .262  329 

to\J 

.032  129 

—  27  610 

3.6 
3.7 

3.8 
3.9 

-  .251  152 
-  .237  825 
-  .222  594 
-  .205  705 

—  11  177 
-  13  323 
-15  231 

-  16  889 

.057  974 
.081  901 
.103  778 
.123  499 

-25  846 
-23  927 
-21  877 
-19  721 

4.0 

-  .187  407 

—  lo  Z\)o 

.  140  982 

—  17  483 

4.1 

4.2 
4.3 
4.4 

-  .167  947 
-  .147  572 
-  .126  524 
-  .105  038 

—  19  460 
-20  375 
-21  048 
-21  486 

.156  165 
.169  013 
.179  510 
.187  660 

-15  183 
-  12  848 
-  10  497 
-  8  150 

4.5 

-  .083  344 

—  ZL  by4 

.193  491 

-  5  831 

4.6 
4.7 

4.8 
4.9 

-  .061  664 
-  .040  209 
-  .019  179 
+  .001  237 

-21  680 
-21  455 
-21  030 
-20  416 

.197  047 
.198  391 
.197  604 
.194  780 

-  3  556 
-  1  344 

+   787 
2  824 

5.0 

+  0.020  865 

—  iy  o^o 

+  0.190  030 

+  4  750 

APPENDIX 


557 


TABLE   IV 
expl  x  and  expl  (  —  x)  from  0.0  to  5.0 


X 

expl  x 

J 

expl  (-x) 

A 

.0 

+      00 

00 

.1 
.2 
.3 
.4 

+  1.622  813 

+   .821  761  - 
+   .302  669 
.  104  765 

+  1.822  924 
1.222  651 
.905  677 
.702  380 

AX.A  99fl 

CCQ  77/1 

.6 

.7 
.8 
.9 

-   .769  881 
-  1.064  907 
-  1.347  397 
-  1.622  812 

.454  380 
.373  769 
.310  597 
.260  184 

i  f\ 

i  on  c  1  1  Q 

91  Q  ^84. 

1.1 
1.2 
1.3 
1.4 

-  2.167  378 
-  2.442  092 
-  2.721  399 
-  3.007  207 

272  260 
274  714 
279  306 

285  809 

9Q/L  O7S 

.  185  991 
.158  408 
.135  451 
.116  219 

oo  oyo 
27  583 
22  957 
19  232 

1  ft  1  QQ 

1.5 

-  3.301  285 

zy<±  u/o 

.  100  020 

ID  iyy 

1.6 
1.7 

1.8 
1.9 

-  3.605  320 
-  3.920  963 
-  4.249  868 
-  4.593  714 

304  034 
315  643 

328  904 
343  846 

.086  308 
.074  655 
.064  713 
.056  204 

13  712 
11  653 
9  942 
8  509 

2  0 

4  954  234 

ooU  OZL 

048  901 

olio 

070  nni 

69S7 

2.1 

2  2 
2~3 
2.4 

-  5.333  235 
-  5.732  615 
-  6.154  381 
-  6.600  670 

399  379 
421  766 
446  289 

A7Q  flQA 

.042  614 
.037  191 
.032  502 
.028  440 

Zo< 
5423 
4  689 
4  062 

3FQr 

2.5 

-  7.073  766 

4/O  U»O 

.024  915 

oZo 

2.6 

2.7 
2.8 
2.9 

-  7.576  115 
-  8.110  347 
-  8:679  298 
-  9.286  024 

502  349 
534  233 
568  950 
606  726 

.021  850 
.019  182 
.016  855 
.014824 

3  065 
2  668 
2  327 
2  031 

3  0 

—  9  933  833 

647  808 

013  048 

776 

3.1 
3.2 
3.3 
3.4 

-10.626  300 
-11.367  303 
-12  161  041 
-13.012  075 

oyz  4oo 
741  002 
793  739 
851  034 

.011  494 
.010  133 
.008  939 
.007  891 

OO4 

1  361 
1  194 
1  048 

Q91 

3.5 

13  925  354 

yio  z/y 

.006  970 

yji 

3.6 
3.7 

3.8 
3.9 

-14.906  254 
-15.960  619 
-17.094  802 
-18.315  714 

980  900 
1.054  365 
1.134  183 
1.220  912 

.006  160 
.005  448 
.004  820 
.004  267 

810 
712 
628 
553 

4  0 

—  19  630  874 

.  olo  loU 

003  779 

48» 

1X17  CQO 

jorv 

4.1 
4.2 
4.3 
4.4 

-21.048  467 
-22.577  401 
-24.227  380 
-26.008  973 

.'I  I/  O\)Z 

1.528  934 
1.649  979 
1.781  593 

1Q94.  79*} 

.003  349 
.002  969 
.002  633 
.002  336 

4oU 

380 
336 
297 

4  5 

—  27  933  697 

002  073 

Juo 

4.6 

4.7 
4.8 
4.9 

-30.014  099 
-32.263  860 
-34.697  890 
-37.332  451 

.  UoU  4Uo 

2.249  760 
2.434  030 
2.634  561 

20C9  QQ  r 

.001  841 
.001  635 
.001  453 
.001  291 

Z6Z 
206 
182 
162 

5.0 

-40.185  275 

.00^  o^o 

+  0.001  148 

14o 

.558 


APPENDIX 


TABLE  V 

Col  x,  sil  x,  expl  x,  and  expl  (  —  x)  from  0  to  15 


X 

col  x 

sil  x 

expl  x 

expl  (-x} 

x 

x° 

0 

+  1  .  570796 

GO 

00 

00 

0 

0 

1 
2 
3 
4 

+  .624713 
-  .034617 
-  .277856 
-  .187407 

-.337404 
-  .422981 
-.119630 
+  .140982 

1.895118 
4.954234 
9.933833 
19.630874 

+  .219384 
.048901 
.013048 
.003779 

1 

2 
3 
4 

57.2958 
114.5916 
171.8874 
229  .  1832 

5 

+  .020865 

+  .190093 

40.185275 

.001148 

5 

286.479 

6 

7 
8 
9 

+  .146109 
+  .116200 
-  .003391 
-  .094244 

+  .068057 
-.076695 
-  .  122434 
-.055348 

85.990 
191  .  505 
440  .  380 
-  1037.878 

.000360082 
.000115482 
.000  37666 
.000012447 

6 

7 
8 
9 

343.775 
401.071 
458.366 
515.662 

10 

-  .087551 

+  .045456 

2492.229 

.000004157 

10 

572.958 

11 

12 
13 
14 

-  .007511 
+  .065825 
+  .071435 
+  .014585 

+  .089563 
+  .049780 
-.026764 
-.069396 

-  6071.406 
-  14959 
-  37198 
-  93193 

.000001400 
.000000475 
.000000162 
.000000056 

11 
12 
13 
14 

630.254 
687.550 
744  .  846 
802  .  142 

15 

-  .047398 

-.046279 

-234956 

+  .000000019 

15 

859.438 

APPENDIX 


559 


TABLE  VI 

col  x  and  sil  x 


X 

co'  x 

sil  x 

x 

col  x 

sil  x 

0 

+  1.570  796 

GO 

150 

+  .004  6.0 

+  .004  796 

5 
10 
15 
20 
25 
30 
35 
40 
45 

+  .020  865 
-  .087  551 
-  .047  398 
+  .022  555 
+  .039  314 
+  .004  040 
-  .026  126 
-  .016  189 
+  .012  081 

+  .  190  093 
+  .045  456 
-.046  279 
-.044  420 
+  .006  849 
+  .033  032 
+  .011  480 
-.019  020 
-.018  632 

160 
170 
180 
190 
200 
300 
400 
500 
600 
700 

-.006  089 
+  .005  529 
-.003  349 
+  .000  377 
+  .002  414 
-  .000  085 
-  .001  319 
-  .001  770 
-.001  665 
001  1  Q8 

-.001  409 
-.002  006 
+  .004  432 
-.005  249 
+  .004  378 
+  .003  332 
+  .002  124 
+  .000  932 
-  .000  076 

OOO  77Q 

50 

+  .019  179 

+  .005  628 

800 
QOO 

-.000  559 
_i_  000  07^ 

-.001  118 

001  1OQ 

cc 

_i_   000  079 

4-  01  8  173 

60 

cc 

-  .015  949 
oofi  fi?  ^ 

+  .004  813 

012  847 

i  ooo 

+  .000  563 

-.000  826 

70 
75 
80 
85 
90 
95 

+  .009  201 
+  .012  217 
-  .001  535 
-  .011  602 
-  .004  867 
+  .007  760 

-.010  922 
+  .005  332 
+  .012  402 
+  .001  935 
-.009  986 
-.007  110 

2  000 
3  000 
4  000 
5  000 
6  000 
7  000 
c  000 

-.000  183 
-.000  325 
-.000  182 
+  .000  031 
+  .000  151 
+  .000  123 
4-  000  008 

-  .000  465 
-.000  073 
+  .000  171 
+  .000  198 
+  .000  071 
-.000  072 

000  19^ 

100 

+  .008  571 

+  .005  149 

9  000 

-.000  088 

-.000  068 

110 

1  Of) 

-  .009  084 

+   OO7  894. 

+  .000  320 

004-  781 

10  000 

-.000  095 

+  .000  031 

130 
140 

-  .002  880 
-  .001  363 

+  .007  132 
-.007  Oil 

11  000 
100  000 
1  000  000 

-.000  026 
-.000  010 
+  000  001 

+  .000  087 
-.000  000 
4-  000  000 

150 

+  .004  630 

+  .004  796 

560 


APPENDIX 
TABLE  VII 

7T'  7t 

MAXIMA  AND  MINIMA  OF  col  —x  and  sil  —x 


X 

col-^c 

2      .  Tt 

col—  re 

7CX       2 

x 

i*; 

2     ..  7T 

•  sil  —x 

TtX       2 

0 
2 

+  1.570  796 
-  .281  141 

1 
3 

-  .472  001 
+  .198  408 

4 

+  .152  645 

5 

-  .123  772 

6 

8 

—  .103  966 

+  .078  635 

7 
9 

+  .089  564 
-  .070  065 

x  +  l 

10 

-  .063  168 

494 

11 

+  .057  501 

374 

12 

+  .052  762 

290 

13 

-  .048  742 

229 

14 

-  .045  289 

184 

15 

+  .042  292 

149 

16 

18 

+  .039  665 
-  .035  281 

123 

94 

17 

19 

-  .037  345 
+  .033  432 

103 
74 

20 

+  .031  767 

64 

21 

-  .030  260 

55 

22 

-  .028  889 

48 

23 

+  .027  637 

42 

24 

+  .026  489 

37 

25 

-  .025  432 

33 

26 

28 

—  .024  456 
+  .022  713 

29 
23 

27 
29 

+  .023  552 
-  .021  931 

26 
21 

30 

-  .021  202 

19 

31 

+  .020  519 

17 

32 

+  .019  879 

15 

33 

-  .019  277 

14 

34 

-  .018  711 

13 

35 

+  .018  177 

12 

36 

38 

.Ull  D/o 

-  .016  744 

: 

10 

37 

39 

-  .017  196 
+  .016  315 

10 
9 

40 

+  .015  907 

9 

41 

-  .015  520 

8 

42 

-  .015  151 

8 

43 

+  .014  799 

7 

44 

+  .014  462 

7 

45 

--  .014  141 

6 

46 

48 

—  .013  834 
+  .013  258 

6 
5 

47 
49 

+  .013  540 
-  .012  988 

5 

4 

50 

-  .012  728 

4 

51 

+  .012  480 

3 

52 

+  .012  239 

3 

53 

-  .012  008 

3 

54 

-  .011  786 

3 

55 

+  .011  572 

3 

56 

58 

.UII  ODD 

-  .010  974 

2 

57 
59 

-  .011  166 
+  .010  788 

3 

2 

60 

+  .010  608 

2 

61 

-  .010  434 

2 

62 

-  .010  266 

2 

63 

+  .010  103 

2 

64 

+  .009  945 

2 

65 

-  .009  792 

2 

66 

68 

—  .009  644 
+  .009  360 

2 

67 
69 

+  .009  500 
-  009  225 

2 
2 

70 

-  .009  093 

2 

71 

+  .008  965 

1 

72 

+  .008  841 

1 

73 

-  .008  719 

1 

74 

7fi 

-  .008  602 

+   nno  07  c 

1 

75 

+  .008  487 

1 

<D 

78 

.  UUo  o/O 

-  .008  161 

. 
1 

77 
79 

-  .008  267 
+  .008  057 

1 
1 

80 

+  .007  957 

1 

81 

-  .007  858 

0 

x> 

80 

j»fe 

79 

X 

sil|s=(-l) 

+  1 
2~2 

INDEX 

PAGE 

Acceleration  constant  of  traveling  wave 466 

Air  blast,  action  in  oscillating-current  generator 75 

pressure  required  in  oscillating-current  generator 75 

Alternating-current  circuit  and  transient  term  of  fundamental  frequency  473 

distribution  in  conductor 369 

transformer  operating  oscillating-current  generator 87 

transmission,  equations  of  traveling  wave 477 

wave  as  traveling  wave  without  attenuation 472 

Alternator  control  by  periodic  transient  term  of  field  excitation 223 

Aluminum  cell  rectifier 222 

effective  penetration  of  alternating  current 378 

Amplitude  of  traveling  wave 465 

of  wave 438 

Arc,  and  spark 249 

continuity  at  cathode 249 

lamp,  control  by  inductive  shunt  to  operating  mechanism 131 

machine 230 

as  rectifier 221 

current  control 220 

properties 249 

rectification 249 

rectifiers 222 

resistivities 9 

starting 249 

Arcing  ground  on  lines  and  cables,  as  periodic  transient  phenomenon  . .  23 

Armature  reactance,  reaction  and  short-circuit  current  of  alternator 199 

Attenuation  of  alternating  magnetic  flux  in  iron 361 

constants 434 

of  traveling  wave,  and  loading 462 

Booster,  response  to  change  of  load 158 

Brush  arc  machine 221,  230,  242,  248 

Building  up  of  direct-current  generator 32 

of  overcompounded  direct-current  machine 49 

Cable,  high-potential  underground,  standing  waves 452 

opening  under  load 112,  118 

short-circuit  oscillation 113,  118 

starting 111,117 

transient  terms  and  oscillations 98,  102 

561 


562  INDEX 

PAGE 

Capacity,  also  see  Condenser. 

and  inductance,  equations 48 

and  velocity  of  propagation 400,  401 

distributed  series 348 

energy  of  complex  circuit 517 

in  mutual  inductive  circuit 161 

of  electric  circuit 112 

range  in  electric  circuit 13 

representing  electrostatic  component  of  electric  field 5 

shunting  direct-current  circuit 133 

specific,  numerical  values 11 

suppressing  pulsations  in  direct-current  circuit 134 

Cast  iron,  effective  penetration  of  alternating  current 378 

Cathode  of  arcs 249 

Charge  of  condenser 51 

of  magnetic  field 27 

Circuit,  complex,  see  Complex  circuit. 

control  by  periodic  transient  phenomena 220,  223 

electric,  speed  of  propagation  in 422 

Closed  circuit  transmission  line 306 

Col  al 392,  394 

Commutation  and  rectification 222 

as  transient  phenomenon 40 

Commutator,  rectifying  229 

Complex  circuit,  of  waves 498 

power  and  energy 513 

resultant  time  decrement 504 

traveling  wave 468 

Compound  wave  at  transition  point 532 

Condenser,  also  see  Capacity. 

charge,  inductive 18 

noninductive 18 

circuit  of  negligible  inductance 55 

equations 48 

oscillation,  effective  value  of  voltage,  current  and  power. ...  70 

efficiency,  decrement  and  output 72 

frequency 62 

general  equations 60 

size  and  rating 69 

starting  on  alternating  voltage 94 

voltage  in  inductive  circuit 49 

Conductance,  shunted,  effective 12 

Conductors  at  high  frequency 403 

Constant-current  mercury  arc  rectifier 250 

rectification 221,  230 

potential-constant-current  transformation  by  quarter-wave  line  308 

mercury  arc  rectifier 251 

rectification 221,  230 


INDEX  563 

PAGE 

Control  of  circuits  by  periodic  transient  phenomena 220 

Conversion  by  quarter-wave  circuits 313 

Copper,  effective  penetration  of  alternating  currents 378 

ribbon,  effective  high  frequency  impedance 408 

wire,  effective  high  frequency  impedance 408 

Cosine  wave,  traveling 434 

Critical  case  of  condenser  charge  and  discharge 53 

resistance  of  condenser  oscillation 66 

start  of  condenser  on  alternating  voltage 95 

Current   density,  in  alternating-current  conductor 372 

effective,  of  oscillating-current  generator 81 

transformation  at  transition  point  of  wave 529 

Damping  of  condenser  oscillation 66,  72 

Decay  of  continuous  current  in  inductive  circuit 17 

of  wave  of  condenser  oscillation 72 

Decrement  of  condenser  oscillation 65,  72 

resultant  time,  of  complex  circuit 504 

Destructive  voltages  in  cables  and  transmission  lines 120 

Dielectric  constant,  numerical  values 11 

strength,  numerical  values 11 

Dielectric  also  see  Electrostatic. 

Direct-current  generator,  self-excitation 32 

railway,  transient  effective  resistance 379 

Disappearance  of  transient  term  in  alternating-current  circuit 43 

Discharge  of  condenser ,  .  51 

Geissler  tube 9 

inductive,  as  wave 535 

into  transmission  line '. 542 

of  motor  field 29 

Disruptive  strength,  numerical  values 11 

voltage  in  opening  direct-current  circuit 26 

Distance  attenuation  constant 434 

in  velocity  measure .  435 

Distortionless  circuit 441,  447 

Distributed  series  capacity 348 

Distribution   of  alternating-current  density  in  conductor 369 

of  alternating  magnetic  flux  in  iron 355 

Divided  circuit,  general  equations 122 

continuous-current  circuit  without  capacity 126 

Dynamostatic  machine 220 

Effective  current  of  condenser  discharge 70 

voltage  and  power  of  oscillating-current  generator.  .  .  81 

layer  of  alternating-current  conductor 377 

penetration  of  alternating  current  in  conductor 376,  378 

power  of  complex  circuit 514 

of  cpndenser  oscillation '70 

reactance  of  armature  reaction 200 


561  INDEX 

PAGE 

Effective  resistance  of  alternating-current  distribution  in  conductor,    370,  376 

voltage  of  condenser  oscillation 70 

Efficiency  of  condenser  oscillation 72 

Electric  circuit,  general  equations 428 

field,  velocity  of  propagation 387 

Electrolytes,  resistivities 8 

Electrolytic  rectifiers 222 

Electromagnetic,  also  see  Magnetic. 

axis  of  electric  field 4 

Electrostatic,    also  see  Dielectric. 

axis  of  electric  field 4 

energy  of  complex  circuit 517 

field,  energy  of 7 

Elimination  of  pulsations  in  direct-current  circuit  by  capacity 134 

Energy  of   complex  circuit 513 

condenser  discharge 70 

electric  field 4,  7 

transfer  in  complex  circuit 507,  521 

constant  of  complex  circuit 507 

Equations,  general,  of  electric  circuit 428 

Even  harmonics  of  half-wave  oscillations 483 

Excitation  of  motor  field 27 

Exponential  curve  of  starting  current 45 

Field  current  at  armature  short-circuit .  t 202 

electric,  of  conductor 3 

energy  of 4 

velocity  of  propagation 387 

excitation,  transient  term 27 

exciting  current,  rise  and  decay 17 

regulation  of  generator  by  periodic  transient  terms 223 

resultant  polyphase   , 192 

Flat  conductor,  unequal  current  distribution   371 

Floating  system  of  control 220 

Fluctuations  of  current  in  divided  circuit 129 

voltage  of  direct-current  generator  with  load 149 

Free  oscillations 432,  478 

and  standing  waves 482 

Frequency,   absence  of  effect  on  circuit  oscillation 10 

and  starting  current  of  transformer 182 

constant  of  wave 433 

limit  of  condenser  oscillation 73 

of  condenser  oscillation 62,  68 

field  current  at  armature  short-circuit 203 

oscillation  of  condenser,  transmission  line,  cable. ...       99,  338 

recurrence  of  discharge  in  oscillating-current  generator. .  81 

wave 433 

range  of  condenser  oscillation 71 

electromagnetic  induction 67 


INDEX  565 

PAGE 

Full-wave  oscillation  of  complex  circuit 508 

transmission  line 330 

Fundamental  frequency  of  oscillation,  cables  and  transmission  lines     103,  105 

Gas  pipe,  effective  high  frequency  impedance 408 

General  circuits  with  inductance  and  capacity 174 

without  capacity 168 

equations  of  electric  circuit 428 

Generator,  direct-current  over-compounded,  building  up 149 

self-excitation 32 

oscillating  current 74 

German  silver,  effective  penetration  of  alternating  current 378 

Gradual  approach  to  permanent  value 21 

or  logarithmic  condenser  charge  and  discharge 53 

term,  also  see  Logarithmic. 

Graphite,  effective  penetration  of  alternating  current 378 

Grounded  transmission  line 303 


Half-wave  oscillation 483,  490 

of  complex  circuit 509 

transmission  line 333 

rectification 221 

Harmonics,  even,  of  half-wave  oscillation 483 

Hertzian  oscillators 388 

High  frequency    alternators,  momentary  short-circuit  current 201 

conductor 370,  403 

discharge 388 

oscillating  currents  by  periodic  transient  terms 220 

oscillations  of  cables  and  transmission  lines 103,  105 

power  surge  of  low-frequency 105 

stray  field  and  starting  current  of  transformer 189 

Impact  angle  at  transition  point  of  wave 527 

Impedance  of  conductor  at  high  frequency 407 

effective  high  frequency 408,  413 

of  radiation 396 

traveling  wave 460 

Inductance  and  shunted  capacity  suppressing  pulsations  in  direct-cur- 
rent circuit 134 

effective,  of  radiation 394 

energy  of  complex  circuit 515 

in  telephone  lines 455,  462 

massed,  in  circuit 537 

of  conductor  without  return 390 

electric  circuit 12 

range  in  electric  circuit 13 

representing  magnetic  component  of  electric  field 5 


566  INDEX 

PAGE 

Induction,  magnetic,  and  starting  current  of  transformer 180 

motor  circuit,  starting 44 

Inductive  discharges  into  transmission  line 542 

shunt  to  non-inductive  circuit 129 

Inductorium,  equations 164 

Infinitely  long  conductor 305 

Input,  see  Power. 

Instantaneous  power  in  complex  circuit 514 

Insulators,  resistivities 9 

Iron  arc  operated  by  oscillating-current  generator 82 

effective  penetration  of  alternating  current 348 

wire,  current  distribution 370 

and  pipe,  effective  high  frequency  impedance 408 

Laminated  iron,  alternating  magnetic  flux 355 

pole  series  booster,  response  to  voltage  change 158 

Layer,  effective,  of  alternating-current  conductor 377 

Leakage  in  telephone  lines 455,  463 

Length  of  wave 433 

Lighting  circuit,  starting 27,  44 

Lightning  arrester,  multigap 348 

conductors 370 

discharges 388 

in  thunder  cloud 350 

Limit  condition  of  condenser  equations 50 

of  frequency  of  condenser  oscillations 73 

Loading  of  telephone  lines 455,  462 

Local  oscillations  of  cables  and  lines 103,  105 

Logarithmic  decrement  of  con^jnser  oscillation 65 

or  gradual  condenser  charge  and  discharge 53 

start  of  condenser  on  alternating  voltage 95 

Low  frequency  surge  in  cables  and  lines 103,  105 

stray  field  and  starting  current  of  transformer 189 

Loxodromic  spiral  of  starting  current 46 

Magnetic,  also  see  Electromagnetic. 

density  and  starting  current  of  transformer 180 

energy  of  complex  circuit 515 

field,  energy  of 6 

flux,  alternating,  in  iron 355,  361,  363,  365,  366,  367 

materials 10 

saturation,  numerical  values 10 

Magnet  poles,  solid,  as  mutual  inductive  circuit 155 

Main  axes  of  electric  field 46 

wave  at  transition  point 531 

Massed  inductance  and  electric  wave 537 

Mechanical  rectification 221,  229 

Mercury  arc  rectifier 250 


INDEX  567 

PAGE 

Metallic  conductors,  resistivities 8 

magnetic  induction 10 

Minimum  wave  length  of  oscillating  currents 74 

Motor  circuit,  alternating,  starting 44 

field,  excitation 27 

Multigap  lightning  arrester '. 348 

Mutual  impedance  and  velocity  of  propagation 399 

inductance,  equations 143 

and  velocity  of  propagation 397 

inductive  circuit  with  capacity 161 

without  capacity 144 

of  solid  magnet  poles 155 

reactance : 143 

Nominal  generated  e.m.f.  and  short-circuit  current 200 

Noninductive   condenser  circuit 54 

shunt  to  inductive  circuit 129 

Nonoscillatory,  see  Gradual  or  Logarithmic. 

Open-circuit  rectification 230 

Opening  of  cable  or  transmission  line  under  load 112,  118 

of  continuous-current  circuit 26 

Open  transmission  line 299 

Oscillating-current  generator 69,  74 

and  charging  current 85 

high-frequency  currents  by  periodic  transient  terms 220 

Oscillation,   also  see  Condenser  discharge. 

free,  of  circuit 432,  478 

of  rotating  field  in  starting 197 

transmission  line 322,  333,  336,  338 

Oscillatory  approach  to  permanent  value 21 

case  of  alternating  circuit 93 

or  trigonometric  condenser  charge  and  discharge 53 

start  of  condenser  on  alternating  voltage 95 

Oscillograms  of  mercury  arc  rectifier 264 

of  transformer  starting  current 190 

Output,  also  see  Power. 

effective,  of  oscillating-current  generator 81 

Overcompounded  direct-current  generator,  building  up 149 

Overlap  of  rectifying  arcs 251 

Overreaching  of  condenser  charge 19 

in  noninductive  branch  of  inductive  circuit 131 

Oversaturated  transformer  flux  of  starting  current 181 

Penetration,  effective  depth  of, 405 

of  alternating  current  in  conductor 376,  378 

magnetic  flux  in  iron 361,  363,  365 

Period   of  recurrence 218 

wave . .  433 


568  INDEX 

PAGE 

Periodic  transient  terms 22,  218 

Permanent  term  of  alternating-current  circuit 91 

values  of  electric  quantities 16 

Permeability 9 

apparent,,  of  iron  for  alternating  currents 355,  367 

Phase  difference  in  transmission  line 296 

of  wave  and  transient  term 45,  91 

Physical  meaning  of  transient  term 103 

Polyphase  alternator  short  circuit 202,  204 

m.m.f .,  resultant 192 

rectification 230 

Potential  regulation  by  periodic  transient  terms 223 

Power  of  complex  circuit 513 

component  of  high  frequency  impedance 412 

gradient  of  electric  circuit 3 

electric  field 48 

output,  effective,  of  oscillating-current  generator 81 

radiated  by  conductor 394,  397 

transfer  in  complex  circuit 521 

constant  of  complex  circuit 507 

Propagation  constant  of  wave 441 

speed  of,  of  wave 422 

field 387 

Pulsation  of  rotating  field  in  starting 197 

Pyroelectrolytes,  resistivities 9 

Quarter-phase  rectification 230 

Quarter-wave  circuit 313 

oscillation 483,  489 

of  complex  circuit 509 

transmission  line 322 

transformer 312 

transmission  line 306,  315 

Quartic  equation  of  divided  circuit 126 

Radiation,  effective  resistance 393 

power  of  conductor 393,  397 

resistance  of  conductor 403 

Rail,  effective  penetration  of  alternating  current 378 

return  of  single-phase  system 370 

transient  effective  resistance 379 

Railway,   direct-current,  transient  rail  resistance 379 

motor,  self-excitation  as  generator 38 

single-phase,  rail  return 370 

Rating  of  capacity  and  inductance 122 

resistors,  reactors,  condensers 69 

Ratio  of  currents,1  oscillating-current  generator 82 

Reactance,  effective,  of  external  field  . .  406 


INDEX  569 

PAGE 

Reactance,  effective  (continued), 

of  internal  field 406 

of  armature  reaction 200 

mutual  inductive 143 

Reaction,  armature,  and  short-circuit  current 199 

Reactor,  size  and  rating 69 

Rectification,    and  commutation 222 

arc 249 

by  periodic  transient  terms 22,  221 

constant-current 230,  242 

potential 230,  236 

mechanical 229 

open-circuit 230 

polyphase 230 

quarter-phase 230,  242 

reversal  or  change  of  circuit  connections 221 

short-circuit 230 

single-phase 229,  231,  236 

Rectifier,  mercury  arc 250 

oscillograms 264 

Rectifying  commutator 222,  229 

Recurrent  transient  terms 218 

Reflected  waves 431 

at  transition  point 527,  531 

Reflection  angle  at  transition  point 527 

of  wave 525 

Refraction  law  of  wave 534 

of  wave 525 

ratio  at  transition  point 534 

Regulation  of  potential  by  periodic  transient  terms . 223 

Remanent  magnetism  in  starting  transformer 181 

Resistance,    and  starting  current  of  transformer 185 

effective,  of  alternating-current  conductor 370,  376 

of  electric  circuit 12 

radiation,  of  conductor 393,  403 

range  in  electric  circuits 13 

representing  power  gradient  of  electric  field 6 

specific,  see  Resistivity. 

thermal,  of  conductor 403 

Resistivity,  numerical  values 8 

Resistor,  size  and  rating 69 

Resonators,  Hertzian 388 

Resultant  polyphase  m.m.f 192 

time  decrement  of  complex  circuit 504 

Rise  of  continuous  current  in  inductive  circuit 17 

voltage  by  transformation  at  transition  point 529 

Rotating  field,  polyphase 192 

Ruhmkorff  coil,  equations 164 


570  INDEX 

PAGE 

Salt  solution,  effective  penetration  of  alternating  current 378 

Saturation,  magnetic,  numerical  values 10 

and  transformer  starting  current 180 

Screening  effect  of  alternating  currents 378 

Self-excitation  of  direct-current  generator 32 

railway  motor  as  generator 38 

series  generator 38 

shunt  generator 37 

Self-inductance  in  direct-current  circuits 26 

and  short-circuit  current  of  alternator 199 

Series  capacity,  distributed 348 

generator,  self-excitation 38 

motor,  self-excitation  as  generator 38 

Short-circuit  current  of  alternator 199,  201 

oscillation  of  cables  and  lines 113,  118 

rectification 229 

Shunt  generator,  self-excitation 32 

motor  field  excitation,  change 27 

Sil  al 392,  394 

Silicon,  effective  penetration  of  alternating  current 378 

Sine  wave  traveling 434 

Single-phase  alternator  short-circuit 202 

railway  rail  return 370 

rectification 229 

Solid  magnet  poles  as  mutual  inductive  circuit 155,  158 

Space  induction,  magnetic 10 

Spark  discharge  in  cables  and  lines  as  periodic  transient  phenomenon.  .       23 

of  condenser  as  periodic  transient  phenomenon 22 

Speed,    effect  on  discharge  of  motor  field 29 

propagation  in  electric  circuit 422 

of  alternating  magnetic  flux  in  iron 366 

Standing  waves 439,  442 

and  free  oscillations 482 

Starting  current  of  transformer,  calculation 188 

and  frequency 182 

magnetic  saturation 180 

remanent  magnetism 181 

resistance 185 

stray  field 184 

oscillograms 190 

of  alternating  current 43 

continuous  current 27 

polyphase  or  rotating  field 192,  197 

oscillation  of  cables  and  lines Ill,  117 

Static,  see  Electrostatic. 

phenomena 13,  105 

Stationary  waves 439,  442 

Steel,  effective  penetration  of  alternating  current 378 


INDEX  571 

PAGE 

Stored  energy  of  complex  circuit 515 

Stranded  conductor,  effective  resistance  of  current  distribution 370 

Stray  field  and  starting  current  of  transformer 184 

Suppression  of  pulsations  of  direct  current  by  capacity  and  inductance.  134 

Synchronous  reactance  and  short-circuit  current 200 

rectifier 221 

Telegraph,  wireless 388 

cable,  submarine,  standing  waves 454 

Telephone 281 

circuit,  long  distance,  standing  waves 454 

Terminal  conditions  of  condenser  equations 50 

Tesla  transformer  and  oscillating-current  generator 82 

Thermal  resistance  of  conductor 403 

Third  harmonic  of  short-circuit  current 213 

Thomson  arc  machine 221,  230 

Thunder  cloud,  lightning  discharge  in 350 

Time  attenuation  constant 434 

constant,  resultant,  of  complex  circuit 504 

decrement,  resultant,  of  complex  circuit 504 

local,  of  traveling  wave 460 

Tirrill  regulator 223 

Transfer  constant  of  energy,  of  complex  circuit 507 

of  energy  in  oscillation  of  complex  circuit 507,  521 

Transformation  ratio  at  transition  point  of  wave 529 

of  voltage  and  current  at  transition  point 529 

Transformer,  alternating,  operating  oscillating-current  generator 87 

distributed  capacity 342 

quarter-wave  oscillation 312 

starting 44 

and  magnetic  saturation 180 

Transient  rail  resistance  with  direct  current 386 

terms,  conditions  of  their  appearance 21,  23 

of  alternating-current  circuit 91 

capacity  and  inductance,  physical  meaning 103 

fundamental  frequency  in  alternating-current  circuit  473 

periodic 22 

unequal  current  distribution 379 

Transition  period  at  change  of  circuit  condition 16 

points  of  wave 498 

Transmission  angle  at  transition  point 527 

line,   characteristic  curves 297 

closed 306 

constants 282 

conversion  by 308,  313 

equations 284,  287 

approximate 294 

free  oscillation .  .  322 


572  INDEX 

PAGE 

Transmission  line  (continued), 

frequency 100,  280,  320,  338 

full-wave  oscillation 336 

general  equations  of  standing  waves 449,  452 

grounded 303 

half-wave  oscillation 333 

inductive  discharges 542 

infinitely  long 305 

natural  period 280,  320 

open 299 

opening  under  load 112,  118 

phase  difference 296 

quarter-wave 306,  313,  315 

oscillation 322 

radiation 283 

resonance  frequency 279 

with  higher  harmonics 280 

short-circuit  oscillation 113,  118 

starting    111,117 

transient  terms  and  oscillations     98,  102 

Transmitted  wave  at  transition  point 527,  531 

Traveling  sine  and  cosine  waves 434 

waves,  general  equations 458 

without  attenuation,  as  alternating  waves 472 

Trigonometric  or  oscillatory  condenser  charge  and  discharge 53 

term,  see  Oscillatory. 

Turbo  alternators,  short-circuit  current 201 

Unequal  alternating-current  distribution  in  conductor 370 

transient-current  distribution  in  conductor , 379 

Velocity,  also  see  Speed. 

measure  o.f  distance 435 

of  propagation  of  electric  field 387 

Voltage   control  by  transient  terms 223 

drop,  high  frequency  conductor 413 

transformation  at  transition  point  of  wave 529 

variation  of  direct-current  generator,  with  load 149 

Voltmeter  across  inductive  circuit,  pulsation 132 

Water,  effective  penetration  of  alternating  current 378 

Wave  of  alternating  magnetism  in  iron 359 

direct  or  main,  and  reflected 431 

length  of  alternating  magnetic  flux  in  iron 361,  365 

constant 434 

minimum,  of  oscillating  current 74 

transmission 281 

Wireless  telegraphy 388 

X-ray  apparatus,  equations 82 


'1  883 


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